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CHAPMAN & HALL/CRC

 AdaptiveMethod of 

Edited by

 A. Vande WouwerPh. Saucez

 W. E. Schiesser

LINES

Boca Raton London New York Washington, D.C.

© 2001 by CRC Press LLC

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This book contains information obtained from authentic and highly regarded sources. Reprinted material

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© 2001 by Chapman & Hall/CRC

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International Standard Book Number 1-58488-231-X

Library of Congress Card Number 00-069347

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Printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

Adaptive method of lines / edited by A. Vande Wouwer, Ph.Saucez, W.E. Schiesser.p. cm.

Includes bibliographical references and index.

ISBN 1-58488-231-X (alk. paper)

1. Differential equations, Partial—Numerical solutions. I. Wouwer, A.Vande (Alain)

II. Saucez, Ph. (Philippe) III. Schiesser, W.E.

QA377 .A294 2001

515′.353—dc21 00-069347

© 2001 by CRC Press LLC

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 Preface

Partial differential equations (PDEs) arise in the mathematical description of a spec-

trum of chemical and physical problems. This broad utility of PDEs is illustrated in

this book with, for example, applications in chemical kinetics, heat and mass transfer,

hydrology, electromagnetism, and astrophysics. The PDE models are usually highly

nonlinear and therefore require numerical analysis and computer-based solution tech-

niques.

The numerical method of lines (MOL) is a comprehensive approach to the solution

of time-dependent PDE problems that basically proceeds in two steps: (1) spatial

derivatives are first approximated using, for example, finite difference or finite el-

ement techniques, and (2) the resulting system of semi-discrete (discrete in space

— continuous in time) ordinary differential equations (ODEs) is integrated in time.The success of this method follows from the availability of high-quality numerical

algorithms and associated software for the solution of stiff systems of ODEs.

Even though most of the ODE solvers automatically adjust the time-step size (and

possibly the order of the integration formula) in order to meet stability and accu-

racy requirements, the conventional MOL proceeds only in a semi-automatic way

since the spatial nodes are held fixed for the entire course of the computation. For

problems developing large spatial transitions, such as steep moving fronts or shocks,

this conventional approach can be inefficient since a large number of uniformly dis-

tributed nodes is required to adequately capture the regions of high spatial activity.Unfortunately, most of the nodes are “wasted” in regions of low spatial activity and

it is therefore desirable to use a procedure that adapts the spatial grid — move or

add/delete nodes — so as to concentrate them in the regions where they are needed,

i.e., to track and accurately resolve important small-scale features. “Adaptive method

of lines” refers to the concept of both temporal and spatial adaptivity in solving

time-dependent PDEs.

The very active MOL community has traditionally shared their algorithms, codes,

and results with others. This book resulted from the joint efforts of a group of authors

who have the privilege of knowing and working with each other.The purpose of this book is threefold:

1. To provide an introduction to the MOL and the concepts of time and space

adaptation

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2. To present a variety of applications from physics and engineering science

3. To describe new methods and codes and to highlight current research

Hence, this book is intended for engineers, physicists, and applied mathematicianswho are not familiar with the MOL, as well as for numerical analysts interested in

recent research results. The book includes several chapters that cover various aspects

of time and space adaptivity in the method of lines.

Chapter 1 is introductory, and surveys the basic concepts of spatial discretization

and time integration in the general MOL formulation. Then, an overview of several

grid adaptation mechanisms is given, including moving grids and grid refinement,

staticanddynamic gridding, theequidistibution principleand theconcept of a monitor

function, the minimization of a functional, and the moving finite element method.

The several methods are illustrated with different test examples from engineering andscience, which show the great diversity of potential applications addressed by the

MOL and adaptive grid techniques.

Chapter 2, titled “Application of the Adaptive Method of Lines to Nonlinear Wave

Propagation Problems,” continues the introduction through a series of modest one-

dimensional (1D) problems solved by an adaptive grid refinement algorithm which

equidistributes an arc-length or a curvature monitor function subject to constraints

on the grid regularity. This algorithm is applied to several model PDEs describing

nonlinear dispersive wave phenomena, including the cubic Schrödinger equation, the

derivative nonlinear Schrödinger equation, the classical Korteweg-de Vries (KdV)equation, the Korteweg-de Vries-Burgers equation, and a fully nonlinear KdV equa-

tion giving rise to compactons. Numerical results for the propagation and the interac-

tion of solitary waves are discussed in terms of computational expense and solution

accuracy.

Chapter 3, titled “Numerical Solutions of the Equal Width Wave Equation Using an

Adaptive Method of Lines,” deals with the equal width (EW) wave equation, which

is a model partial differential equation for the simulation of 1D wave propagation

in media with nonlinear wave steepening and dispersion processes. The background

of the EW equation is reviewed and this equation is solved by using an advancednumerical method of lines with an adaptive grid whose node movement is based on

an equidistribution principle. The solution procedure is described and the perfor-

mance of the solution method is assessed by means of computed solutions and error

measures. Many numerical solutions are presented to illustrate important features

of the propagation of solitary waves, the interaction of inelastic solitary waves, the

inelastic solitary waves, the breakup of a Gaussian pulse into solitary waves, and the

development of an undular bore.

Chapter 4, titled “Adaptive Method of Line for Magnetohydrodynamic PDE Mod-

els,” is devoted to magnetohydrodynamic PDE models, which describe many inter-

esting phenomena from astrophysics. These PDE systems consist of conservation

laws for mass, momentum, total energy, and induction of magnetic fields. Very often,

these models possess solutions having high spatial gradients that also move rapidly in

time, such as steep moving temperature fronts, rotating sharp pulses, or shock waves.

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In this chapter, an adaptive moving grid method is described that can be used to follow

the steep gradients of the solutions in time. The method is based on an equidistribu-

tion principle enhanced with smoothing terms in the spatial and temporal direction.

To follow the MOL approach, the semi-discretized PDE models that are transformed

to a moving frame are coupled to the ODEs for the grid motion. A suitable stiff time-

integrator is needed to obtain the fully discretized numerical solutions. Numerical

results are shown for some interesting test cases: a magnetic shock-tube problem,

a model for the propagation of shear Alfvén waves, and a model that describes the

advection of a current-carrying cylinder.

Chapter 5, titled “Development of a 1D Error-Minimizing Moving Adaptive Grid

Method,” focuses on the design of a discretization technique dedicated to grid adap-

tation. This so-called compatible scheme allows the leading term of the local residual

to be evaluated directly in terms of a local error in the numerical solution (the nu-merical modeling error). An error-dependent smoothing technique is used to ensure

that higher-order error terms are negligible. The numerical modeling error is mini-

mized by means of grid adaptation. Fully converged adapted grids with strong local

refinements are obtained for a steady-state shallow-water application with a hydraulic

 jump. An unsteady application confirms the importance of taking the error in time

into account when adapting the grid in space. The shortcomings of the present im-

plementation and the remedies currently under development are discussed.

Chapter 6, titled “An Adaptive Method of Lines Approach for Modeling Flow

and Transport in Rivers,” considers practical problems in hydrology, which requirethe accurate simulation of flow and/or transport in natural rivers. Three particular

applications are discussed: (1) the forecasting of water levels, (2) the simulation of 

the transport of soluble substances, and (3) a two space-dimensional (2D) calculation

of flood planes. The basic equations for the simulation of flow and transport in rivers

are presented and the method of lines is proposed for their numerical solution. Due to

the hyperbolicity of the flow equations, Godunov-type upwind schemes are applied to

space discretization, whereas time-integration of thesemi-discretized PDEs is done by

a special variant of thewell-knownfourthorder Rosenbrock-Wanner methodRODAS.

Finally, the use of adaptive space meshes for the current problems is discussed andthe efficiency of the proposed numerical solution methods is demonstrated through

some realistic applications.

Chapter 7, titled “An Adaptive Mesh Algorithm for Free Surface Flows in General

Geometries,” devises a numerical method for computing incompressible free surface

flows ingeneral, threespace-dimensional (3D) geometries. Adaptivemesh refinement

as described by Berger and Colella [ J. Comp. Phys., 82, (1989), pp. 64–84] and

Almgren et al. [ J. Comp. Phys., 142, (1998), pp. 1–46] is used. The free surface

separating the gas and liquid is modeled using “embedded boundary” techniques,

which allow for the arbitrary merge and break-up of fluid mass while maintaining

excellentmass conservation. An embedded boundary method is also used to represent

irregulargeometries, e.g., ship hull. Computational results arepresentedfor3Djetting

problems and 3D ship wave problems. In the process of describing the adaptive

Cartesian grid algorithm for incompressible flow, a new (easy) way for enforcing the

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contact angle boundary condition at points where the free surface meets the geometry

is presented.

Chapter 8, titled “The Solution of Steady PDEs on Adjustable Meshes in Multi-

dimensions Using Local Descent Methods,” focuses on a variant of the multidimen-sional Moving Finite Element (MFE) method for steady PDEs called Least-Squares

Moving Finite Elements (LSMFE). The MFE method is an adaptive method of lines

approach in which the mesh movement is generated by an extended Galerkin method.

The discovery of an optimal property of the steady MFE equations has recently led

to LSMFE. In addition, the implementation of a local approach to the movement of 

the mesh provides new control in the adjustment of the mesh. This new approach can

also be used in a similar way with other space discretization techniques in multidi-

mensions, for example in finite-dimensional approximations in variational principles,

which can include the least-squares best-fit problem, and least-squares methods for

conservation laws. In the latter case, a link has been shown with the equidistribution

of residuals. In this chapter, the techniques are analyzed and 2D examples, including

the advection equation, a shallow water application with a hydraulic jump and the

Euler equations, are given.

Chapter 9, titled “Linearly Implicit Adaptive Schemes for Singular Reaction-

Diffusion Equations,” is concerned with modified adaptive difference schemes for

solving degenerate nonlinear reaction-diffusion equations with singular source terms.

Differential equation problems play important roles in mathematical models of steady

and unsteady combustion processes. Both semi-adaptive and fully adaptive schemes

for solving the aforementioned problems are discussed. In the former case, an adap-

tive, or moving mesh, mechanism in time is considered, while in the latter, adaptation

both in time and space are constructed. Modified monitor functions based on the arc

length of the rate function ut  are obtained. Properties of the numerical schemes are

analyzed and it is shown that under proper smoothness, consistency, and stepsize con-

straints, the numerical solution preserves the monotonicity of the physical solution.

Numerical experiments with quenching phenomena in reaction-diffusion problems

are given to further demonstrate the monotonicity and convergence properties of the

methods.

Chapter 10, titled “Adaptive Linearly Implicit Methods for Heat and Mass Transfer

Problems,” deals with a combination of linearly implicit time integrators of Rosen-

brock type and adaptive multilevel finite elements based on a posteriori error esti-

mates. In the classical MOL approach, the spatial discretization is done once and

for all. Here, a local spatial refinement is allowed in each time step, which results

in a discretization sequence first in time then in space. The spatial discretization

is considered as a perturbation, which has to be controlled within each time step.

This approach has proven to work quite satisfactorily for a wide range of challenging

practical problems. The performance of the adaptive method is demonstrated for two

applications that arise in the study of flame balls and brine transport in porous media.

Chapter 11, titled “Unstructured Adaptive Mesh MOL Solvers for Atmospheric

Reacting Flow Problems,” discusses the application of the method of lines to reacting

flow problems in combustion and atmospheric dispersion. The chapter describes the

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finite volume spatial discretization methods used and indicates the error estimation

approach employed to guide spatial mesh adaptation. The integration methods em-

ployed in time are extensions of existing MOL codes with careful treatment of the

nonlinear equations which minimizes the computation cost without sacrificing ac-

curacy. Examples from a number of large-scale problems are used to illustrate the

approach employed.

Chapter 12, titled “Two-Dimensional Model of a Reaction-Bonded Aluminum

Oxide Cylinder,” concentrates on a particular chemical engineering application. The

reaction-bonded aluminum oxide process utilizes the oxidation of intensely milled

aluminum and Al2O3 powder compacts that are heat-treated in air to make alumina-

based ceramics. A two-dimensional, simultaneous mass and energy balance model

is developed in cylindrical coordinates to describe this process. The model describes

the propagation of an ignition front that has been observed during reaction-bonding.

The model is solved using the method of lines and spatial remeshing techniques basedon the equidistribution principle and spatial regularization procedures introduced in

Chapters 1 and 2.

Chapter 13, titled “Method of Lines within the Simulation Environment DIVA

for Chemical Processes,” introduces the simulation environment DIVA, which is an

integrated numerical tool for modeling, simulation, analysis, and optimization of 

single chemical process units as well as integrated production plants. Attention is

focused on the symbolic preprocessing tool SYPPROT, which allows an automatic

methodof linesdiscretizationof chemicalprocess models with distributedparameters.

A symbolic model formulation with finite difference and finite volume discretizationschemes on fixed as well as moving spatial grids is provided. These methods allow

a convenient model implementation and a flexible application of the MOL. The use

of the preprocessing tool and the numerical methods in DIVA are illustrated with

application examples from chemical engineering.

In summary, the authors of these chapters have provided an introduction to the

adaptive method of lines, and applications ranging from modest 1D PDEs, to complex

2D and 3D PDE systems. In the process of covering this spectrum of applications,

the authors discuss state-of-the-art numerical algorithms for the adaptive solution of 

PDEs in space and time that produce solutions to difficult PDE problems requiring, inparticular, high spatial resolution. This book evolved from the fruitful collaboration

among the chapter authors, and could not have been achieved without their motivation

and enthusiastic support. In order to continue the development of adaptive methods

for PDEs, the authors welcome inquiries about their work.

Alain Vande Wouwer

Philippe Saucez

William Schiesser

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Contributors

I. Ahmad SSO NSLD (CHASSNUPP), P.O. Box 113, Islamabad, Pakistan,dridreesahamad@yahoo.com

M.J. Baines Department of Mathematics, University of Reading, P.O. Box220, Reading, RG6 6AX, U.K., m.j.baines@reading.ac.uk

M. Berzins School of Computing, The University of Leeds, Leeds LS2 9JT,U.K., martin@comp.leeds.ac.uk

M. Borsboom WF | Delft Hydraulics, Marine Coastal and IndustrialInfrastructure, P.O. Box 177, 2600 MH Delft, The Netherlands,mart.borsboom@wldelft.nl

H.S. Caram Department of Chemical Engineering, Iacocca Hall, 111 Re-search Drive, Lehigh University, Bethlehem, Pennsylvania 18015, U.S.A.,

hsc0@lehigh.edu

H.M. Chan Department of Materials Science and Engineering, WhitakerLaboratory No. 5, Lehigh University, Bethlehem, Pennsylvania 18015,U.S.A., hmc0@lehigh.edu

B. Erdmann Scientific Computing, Konrad-Zuse-Zentrum fürInformationstechnik Berlin, Takustrasse 7, 14195 Berlin-Dahlem,Germany, erdmann@zib.de

S. Ghorai Department of Mathematics, Indian Institute of Technology,Kanpur, Kanpur-208016, India, sghorai@iitk.ac.in

E.D. Gilles Max-Planck-Institute for Dynamics of Complex TechnicalSystems, Leipziger Str. 44, D-39120 Madgeburg, Germany, gilles@mpi-madgeburg.mpg.de

 J.J. Gottlieb Institute for Aerospace Studies, University of Toronto, 4925Duffer in S t reet , Toronto , Ontar io , M3H 5T6, Canada,gottlieb@bach.utias.utoronto.ca

S. Hamdi Institute for Aerospace Studies, University of Toronto, 4925Duffer in S t reet , Toronto , Ontar io , M3H 5T6, Canada,samir.hamdi@utoronto.ca

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 J.S. Hansen Institute for Aerospace Studies, University of Toronto, 4925Duffer in S t reet , Toronto , Ontar io , M3H 5T6, Canada,hansen@bach.utias.utoronto.ca

M.P. Harmer Department of Materials Science and Engineering, WhitakerLaboratory No. 5, Lehigh University, Bethlehem, Pennsylvania 18015,U.S.A., mph2@lehigh.edu

R. Keppens F.O.M. Institute for Plasma Physics ‘Rijnhuizen’, P.O. Box 1207,3430 BE, Nieuwegein, The Netherlands, keppens@rijnh.nl

A.Q. Khaliq Department of Mathematics, Western Illinois University, Ma-comb, Illinois 61455, U.S.A., a-khaliq@wiu.edu

A. Kienle Max-Planck-Institute for Dynamics of Complex TechnicalSystems, Leipziger Str. 44, D-39120 Madgeburg, Germany, kienle@mpi-madgeburg.mpg.de

R. Köhler Institut f ür Systemdynamik und Regelungstechnik, UniversitätS tut tgar t , Pfaf fenwaldr ing 9 , D-70550 S tut tgar t , Germany,koehler@isr.uni-stuttgart.de

 J. Lang Scientific Computing, Konrad-Zuse-Zentrum f ür Informationstech-nik Berlin, Takustrasse 7, 14195 Berlin-Dahlem, Germany, lang@zib.de

M. Mangold Max-Planck-Institute for Dynamics of Complex TechnicalSystems, Leipziger Str. 44, D-39120 Madgeburg, Germany, mangold@mpi-madgeburg.mpg.de

K.D. Mohl Institut f ür Systemdynamik und Regelungstechnik, UniversitätStuttgart, Pfaffenwaldring 9, D-70550 Stuttgart, Germany, mohl@isr.uni-

stuttgart.de

P. Rentrop IWRMM, Universität Karlsruhe (TH), Engesser Str. 6, Kaiserin-Augusta-Anlagen 15-17 , D-76128 Kar ls ruhe , Germany,sekretariat@iwrmm.math.uni-karlsruhe.de

Ph. Saucez Laboratoire de Mathématique et Recherche Operationelle,Fac u l té P ol y t ec h n i q u e d e Mon s , 7000 Mon s , B e l g i u m ,saucez@mathro.fpms.ac.be

W.E. Schiesser Iacocca Hall, 111 Research Drive, Lehigh University, Bethle-hem, Pennsylvania 18015, U.S.A., wes1@lehigh.edu

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H. Schramm Institut f ür Systemdynamik und Regelungstechnik, Univer-sität Stuttgart, Pfaffenwaldring 9, D-70550 Stuttgart, Germany,schramm@isr.uni-stuttgart.de

Q. Sheng Department of Mathematics, University of Louisiana, Lafayette,Louisiana, 70504-1010, U.S.A., qxs2336@louisiana.edu

E. Stein Max-Planck-Institute for Dynamics of Complex Technical Systems,Leipziger Str. 44, D-39120 Madgeburg, Germany, stein@mpi-madgeburg.mpg.de

G. Steinebach Bundesanstalt f ür Gewässerkunde, Kaiserin-Augusta-Anlagen 15-17, D-56068 Koblenz, Germany, steinebach@bafg.de

M. Sussman Department of Mathematics, Florida State University, Talla-hassee, Florida 32306, U.S.A., sussman@math.fsu.edu

A.S. Tomlin Department of Fuel and Energy, , The University of Leeds,Leeds LS2 9JT, U.K., A.Tomlin@chemistry.leeds.ac.uk

 J. Ware 35 Gun Place, 86 Wapping Lane, Wapping, London, U.K., justinware@onetel.net.uk

M.J. Watson Department of Chemical Engineering, Iacocca Hall, 111 Re-search Drive, Lehigh University, Bethlehem, Pennsylvania 18015, U.S.A.,watsonmj@apci.com

A. Vande Wouwer Laboratoire d’Automatique, Faculté Polytechnique deMons, 7000 Mons, Belgium, vdw@autom.fpms.ac.be

P.A. Zegeling Mathematical Institute, Utrecht University, Budapestlaan 6,

3584 CD Utrecht, The Netherlands, zegeling@math.uu.nl

M. Zeitz Institut f ür Systemdynamik und Regelungstechnik, UniversitätStuttgart, Pfaffenwaldring 9, D-70550 Stuttgart, Germany, zeitz@isr.uni-stuttgart.de

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Contents

1 Introduction 

 Alain Vande Wouwer, Philippe Saucez, and William Schiesser 

1.1 Classification of Partial Differential Equations 

1.2 The Method of Lines 

1.2.1 Spatial Discretization

1.2.2 Time Integration 

1.3 Adaptive Grid Methods

1.3.1 Grid Adaptation Criteria 

1.3.2 Static vs. Dynamic Gridding 

1.3.3 Moving Grid and Grid Refinement Algorithms 

1.3.4 Grid Regularity

1.4 Case Studies 

1.4.1 Case Study 1

1.4.2 Case Study 2 

1.4.3 Case Study 3 

1.4.4 Case Study 4 

1.4.5 Case Study 5 

1.4.6 Case Study 61.5 Summary 

References 

2 Application of the Adaptive Method of Lines to Nonlinear Wave

Propagation Problems 

 Alain Vande Wouwer, Philippe Saucez, and William Schiesser 

2.1 Introduction 

2.2 Adaptive Grid Refinement 

2.2.1 Grid Equidistribution with Constraints

2.2.2 Time-Stepping Procedure and Implementation Details 

2.3 Application Examples 

2.3.1 The Nonlinear Schrödinger Equation 

2.3.2 The Derivative Nonlinear Schrödinger Equation 

2.3.3 The Korteweg-de Vries Equation

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2.3.4 The Korteweg-de Vries-Burgers Equation 

2.3.5 KdV-Like Equations: The Compactons 

2.4 Conclusions

References 

3 Numerical Solutions of the Equal Width Wave Equation Using an

Adaptive Method of Lines 

S. Hamdi, J.J. Gottlieb, and J.S. Hansen

3.1 Introduction

3.2 Equal-Width Equation 

3.3 Numerical Solution Procedure 

3.4 Numerical Results and Discussion 

3.4.1 Single Solitary Waves

3.4.2 Inelastic Interaction of Solitary Waves 

3.4.3 Gaussian Pulse Breakup into Solitary Waves 

3.4.4 Formation of an Undular Bore

3.5 Concluding Remarks

References

4 Adaptive Method of Lines for Magnetohydrodynamic PDE Models 

P. A. Zegeling and R. Keppens

4.1 Introduction 

4.2 The Equations of Magnetohydrodynamics 

4.3 Adaptive Grid Simulations for 1D MHD

4.3.1 The MHD Equations in 1D 

4.3.2 The Adaptive Grid Method in One Space Dimension 

4.3.3 Numerical Results 

4.4 Towards 2D MHD Modeling

4.4.1 2D Magnetic Field Evolution 

4.4.2 Adaptive Grids in Two Space Dimensions 

4.5 Conclusions 

References

5 Development of a 1-D Error-Minimizing Moving Adaptive Grid

Method 

 Mart Borsboom

5.1 Introduction 

5.2 Two-Step Numerical Modeling 

5.3 1-D Shallow-Water Equations 

5.4 Compatible Discretization 

5.4.1 Discretized Shallow-Water Equations

5.4.2 Iterative Solution Algorithm

5.5 Error Analysis

5.5.1 Error Analysis in Space 

5.5.2 Error Analysis in Time 

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5.5.3 Error in Discretized Shallow-Water Equations

5.6 Error-Minimizing Grid Adaptation 

5.7 Results

5.7.1 Steady-State Application 

5.7.2 Unsteady Application 

5.8 Conclusions

References

6 An Adaptive Method of Lines Approach for Modeling Flow and

Transport in Rivers 

Gerd Steinebach and Peter Rentrop

6.1 Introduction 

6.2 Modeling Flow and Transport in Rivers 

6.3 Method of Lines Approach

6.3.1 Network Approach 

6.3.2 Space Discretization 

6.3.3 Time Integration 

6.4 Adaptive Space Mesh Strategies 

6.4.1 Extension to 2D Problems 

6.5 Applications

6.6 Conclusion 

References 

7 An Adaptive Mesh Algorithm for Free Surface Flows in General

Geometries 

 Mark Sussman

7.1 Introduction

7.1.1 Overview: Adaptive Gridding 

7.1.2 Overview: Free Surface Model

7.1.3 Overview: Modeling Flows in General Geometries 

7.2 Governing Equations 

7.2.1 Projection Method 

7.3 Discretization

7.3.1 Thickness of the Interface 

7.4 Coupled Level Set Volume of Fluid Advection

7.5 Discretization in General Geometries

7.5.1 Projection Step in General Geometries

7.5.2 Contact-Angle Boundary Condition in General Geometries 

7.5.3 CLS Advection in General Geometries 

7.6 Adaptive Mesh Refinement 

7.6.1 Time-Stepping Procedure for Adaptive Mesh Refinement

7.7 Results and Conclusions 

7.7.1 Axisymmetric Jetting Convergence Study 

7.7.2 3D Ship Waves

References

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8 The Solution of Steady PDEs on Adjustable Meshes in Multidimensions

Using Local Descent Methods 

 M.J. Baines

8.1 Introduction

8.2 Moving Finite Elements 

8.2.1 MFE in the Steady-State Limit

8.2.2 Minimization Principles and Weak Forms 

8.2.3 An Optimal Property of the Steady MFE Equations 

8.3 A Local Approach to Variational Principles 

8.3.1 Descent Methods

8.3.2 A Local Approach to Best Fits 

8.3.3 Direct Optimization Using Minimization Principles 

8.3.4 A Discrete Variational Principle8.4 Least-Squares Methods 

8.4.1 Least-Squares Moving Finite Elements 

8.4.2 Properties of the LSMFE Method 

8.4.3 Minimization of Discrete Norms

8.4.4 Least-Squares Finite Volumes 

8.4.5 Example 

8.5 Conservation Laws by Least Squares 

8.5.1 Use of Degenerate Triangles

8.5.2 Numerical Results for Discontinuous Solutions8.6 Links with Equidistribution 

8.6.1 Approximate Multidimensional Equidistribution 

8.6.2 A Local Approach to Approximate Equidistribution

8.6.3 Approximate Equidistribution and Conservation

8.7 Summary

References

9 Linearly Implicit Adaptive Schemes for Singular Reaction-Diffusion

Equations Q. Sheng and A.Q.M. Khaliq

9.1 Introduction 

9.2 The Semi-Adaptive Algorithm 

9.2.1 The Discretization 

9.2.2 The Adaptive Algorithms 

9.3 The Fully Adaptive Algorithm 

9.3.1 The Discretization

9.3.2 The Monotone Convergence 

9.3.3 The Error Control and Stopping Criterion

9.4 Computational Examples and Conclusions

References

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10 Adaptive Linearly Implicit Methods for Heat and Mass Transfer

Problems 

 J. Lang and B. Erdmann

10.1 Introduction

10.2 Linearly Implicit Methods 

10.3 Multilevel Finite Elements 

10.4 Applications

10.4.1 Stability of Flame Balls

10.4.2 Brine Transport in Porous Media 

10.5 Conclusion 

References 

11 Unstructured Adaptive Mesh MOL Solvers for Atmospheric Reacting-

Flow Problems 

 M. Berzins, A.S. Tomlin, S. Ghorai, I. Ahmad, and J. Ware

11.1 Introduction

11.2 Spatial Discretization and Time Integration 

11.3 Space-Time Error Balancing Control

11.4 Fixed and Adaptive Mesh Solutions 

11.5 Atmospheric Modeling Problem 

11.6 Triangular Finite Volume Space Discretization Method

11.7 Time Integration 

11.8 Mesh Generation and Adaptivity

11.9 Single-Source Pollution Plume Example

11.10Three Space Dimensional Computations 

11.11Three Space Dimensional Discretization 

11.11.1 Flux Evaluation Using Edge-Based Operation 

11.11.2 Adjustments of Wind Field

11.11.3 Advection Scheme

11.11.4 Diffusion Scheme 

11.12Mesh Adaptation11.13Time Integration for 3D Problems 

11.14Three-Dimensional Test Examples

11.14.1 Grid Adaptation 

11.14.2 Downwind Concentration 

11.15Discussions and Conclusions

References

12 Two-Dimensional Model of a Reaction-Bonded Aluminum Oxide

Cylinder 

 M.J. Watson, H.S. Caram, H.M. Chan, M.P. Harmer, Ph. Saucez,

 A. Vande Wouwer, and W.E. Schiesser 

12.1 Introduction 

12.2 Model Development

12.2.1 Model Assumptions 

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12.2.2 Continuum Model Equations 

12.2.3 Initial and Boundary Conditions

12.2.4 Parameters

12.2.5 Dimensionless Equations

12.2.6 Method of Solution

12.3 Results 

12.3.1 Furnace Conditions 

12.3.2 Numerical Solutions 

12.4 Discussion 

12.5 Summary

References

13 Method of Lines within the Simulation Environment Diva for Chemical

Processes 

 R. Köhler, K.D. Mohl, H. Schramm, M. Zeitz, A. Kienle, M. Mangold,

 E. Stein, and E.D. Gilles

13.1 Introduction 

13.2 Architecture of the Simulation Environment Diva

13.2.1 The Diva Simulation Kernel

13.2.2 Code Generation of Diva Simulation Models 

13.2.3 Symbolic Preprocessing Tool 

13.2.4 Computer-Aided Process Modeling 

13.3 MOL Discretization of PDE and IPDE

13.3.1 Finite-Difference Schemes 

13.3.2 Finite-Volume Schemes 

13.3.3 High-Resolution Schemes 

13.3.4 Equidistribution Principle Based Moving Grid Method 

13.4 Symbolic Preprocessing for MOL Discretization

13.4.1 MathematicaData Structure

13.4.2 Procedure of the MOL Discretization

13.5 Application Examples

13.5.1 Circulation-Loop-Reactor Model

13.5.2 Moving-Bed Chromatographic Process

13.6 Conclusions and Perspectives 

References 

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Chapter 1

 Introduction

Alain Vande Wouwer, Philippe Saucez, and William Schiesser

1.1 Classification of Partial Differential Equations

Partial differential equations (PDEs) are one of the most widely used forms of math-

ematics in science and engineering. This is due in large part to the three-dimensional

form of our physical world, and its variation with time. Thus, PDEs have four inde-

pendent variables, that is, three spatial dimensions and time. The variation of physical

properties, e.g., density, velocity, momentum, and energy, is expressed by PDEs in

terms of partial derivatives. For example, if ρ denotes density, then the dependency of 

density on space, x, and time, t , can be denoted as ρ(x, t ), where x is a three-vector (a

vector with three components), and partial derivatives signify the variation of density

with space and time. For example,

∂ρ

∂t  ⇔ρt 

is the first order partial derivative of ρ with respect to t . Note that the partial derivative,∂ρ

∂t  , can also be expressed as a subscripted variable, ρt  .

A PDE that expresses the variation of ρ with x and t  for a fluid, the equation of 

continuity,

∂ρ

∂t = ∇ · (vρ) (1.1)

states a basic physical principle, conservation of mass, where

v fluid velocity vector

∇  divergence operator

∇ is a vector differential operator that has three components in specific coordinate

systems, for example, see Table 1.1.

When working with PDEs, we may also require the gradient of a scalar, see

Table 1.2.

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Table 1.1  ∇· (divergence of a vector )

Coordinate System Components

Cartesian

[∇ ]x = ∂

∂x

[∇ ]y = ∂

∂y

[∇ ]z = ∂

∂z

cylindrical

[∇ ]r = 1

r

∂

∂r(r )

[∇ ]θ  = 1

r

∂

∂θ 

[∇ ]z = ∂

∂z

spherical

[∇ ]r = 1r2

∂∂r

(r2 )

[∇ ]θ  =1

r sin θ 

∂

∂θ (sin θ )

[∇ ]φ = 1

r sin θ 

∂

∂φ

Finally, when working with PDEs, we often require a combination of the two

preceding vector differential operators, i.e., the divergence of the gradient of a scalar,

see Table 1.3.

The derivation of ∇ · ∇ (the Laplacian) follows directly from the preceding com-

ponents of ∇· (divergence of a vector in Table 1.1) and ∇ (gradient of a scalar in

Table 1.2).

Cartesian coordinates:

∇ · ∇ =

i∂

∂x+ j

∂

∂y+ k

∂

∂z

·

i∂

∂x+ j

∂

∂y+ k

∂

∂z

= ∂2

∂x2+ ∂2

∂y 2+ ∂2

∂z2

Cylindrical coordinates:

∇ · ∇ =

ir

1

r

∂

∂r(r) + jθ 

1

r

∂

∂θ + kz

∂

∂z·

ir

∂

∂r+ jθ 

1

r

∂

∂θ + kz

∂

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Table 1.2 ∇ (gradient of a scalar )

Coordinate System Components

Cartesian

[∇ ]x = ∂

∂x

[∇ ]y = ∂

∂y

[∇ ]z = ∂

∂z

cylindrical

[∇ ]r = ∂

∂r

[∇ ]θ  =1

r

∂

∂θ 

[∇ ]z = ∂

∂z

spherical

[∇ ]r = ∂∂r

[∇ ]θ  =1

r

∂

∂θ 

[∇ ]φ = 1

r sin θ 

∂

∂φ

= 1

r

∂

∂r

r

∂

∂r

+ 1

r

∂

∂θ 

1

r

∂

∂θ 

+ ∂

∂z

∂

∂z

= 1

r

∂

∂r+ r

∂2

∂r2

+ 1

r2

∂

∂θ 

∂

∂θ + ∂

∂z

∂

∂z

=

∂2

∂r2+ 1

r

∂

∂r

+ 1

r2

∂2

∂θ 2+ ∂2

∂z2

Spherical coordinates:

∇ · ∇ =

ir1

r2

∂

∂r

r2

+ jθ 1

r sin θ 

∂

∂θ (sin θ  ) + kφ

1

r sin θ 

∂

∂φ

·

ir

∂

∂r+ jθ 

1

r

∂

∂θ + kφ

1

r sin θ 

∂

∂φ

= 1

r2

∂

∂r r2 ∂

∂r+ 1

r sin θ 

∂

∂θ  sin θ 

1

r

∂

∂θ + 1

r sin θ 

∂

∂φ 1

r sin θ 

∂

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Table 1.3  ∇ · ∇ (divergence of the gradient of a scalar )

Coordinate System Component

Cartesian∂2

∂x2+ ∂2

∂y 2+ ∂2

∂z2

cylindrical

∂2

∂r2+ 

1

r

∂

∂r

+ 1

r2

∂2

∂θ 2+ ∂2

∂z2

spherical1

r2

∂

∂r

r2 ∂

∂r

+ 1

r2 sin θ 

∂

∂θ 

sin θ 

∂

∂θ 

+ 1

r2 sin2 θ 

∂2

∂φ2

= 1

r2

∂

∂r

r2 ∂

∂r

+ 1

r2 sin θ 

∂

∂θ 

sin θ 

∂

∂θ 

+ 1

r2 sin2 θ 

∂2

∂φ2

Thus, the equation of continuity in Cartesian coordinates (from (1.1) and Table 1.1)

is:

∂ρ

∂t =

i∂

∂x+ j

∂

∂y+ k

∂

∂z

· ivx ρ + jvy ρ + kvzρ

=∂

∂x (vx ρ) +∂

∂y

vy ρ+

∂

∂z (vzρ)  (1.2)

The term∂

∂x(vx ρ) in (1.2) has a clear physical meaning. The term in parentheses,

(vx ρ), is the mass flux in the x direction. This interpretation is suggested by the units

of this term, e.g.,

(m/s)

kg/m3

= kg/s − m2 .

Thus, (vx ρ) is the kg/s of fluid flowing through a unit area in the x direction (a mass

flux). Consequently,∂

∂x(vx ρ) is the change in this flux with x. We might expect,

intuitively, that this term could undergo a sharp change with respect to x, and this

is indeed the case; that is, (1.2) can propagate sharp changes or fronts, and even

discontinuities, which is the main reason why first-order equations such as (1.2) (note

that it has only first-order derivatives in x and t ) are generally difficult to integrate

numerically. This conclusion suggests that we might benefit from classifying PDEs

as a way of anticipating the general properties of their solutions.

The conventional geometric classification of PDEs as elliptic, hyperbolic, or

 parabolic is expressed through a single, linear PDE. However, this classification

is quite restrictive and therefore not very useful for most applications; we therefore

adopt a less rigorous, but more general, geometrical classification that is illustrated

by Table 1.4.

Note that there are two classes of hyperbolic PDEs, i.e., first and second order. The

two are related. For example, if we define two variables

v = ux , w = ut 

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Table 1.4  Geometric Classification of PDEs with Examples

Order in x (BV) Order in t  (IV) Classification Example

1 1 First-order hyperbolic∂u

∂t = −v

∂u

∂x(advection equation)

2 2 Second-order hyperbolic∂2u

∂t 2= c2 ∂2u

∂x2

(wave equation)

2 1 Parabolic∂u

∂t = α

∂2u

∂x2

(Fourier’s orFick’s second law)

2(in x, y) 0 Elliptic∂2u

∂x2+ ∂2u

∂y 2= 0

(Laplace’s equation)

then by differentiation

vt  = uxt , wx = utx .

If the mixed partial derivatives uxt  and utx are assumed equal, we have

vt  = wx  (1.3)

Also, from the wave equation in Table 1.4,

wt  = c2vx  (1.4)

Thus, the wave equation (a second-order hyperbolic PDE) is expressed in terms of 

two first-order hyperbolic PDEs, (1.3) and (1.4).

Combinations of these classes of PDEs are also possible. For example,

∂u

∂t = −v

∂u

∂x+ D

∂2u

∂x2(1.5)

is hyperbolic-parabolic. The second derivative, D∂2u

∂x2

, which is the x-component of 

the Laplacian in Cartesian coordinates from Table 1.3, generally describes diffusion

(as in Fourier’s second law from Table 1.4). Thus, (1.5) is also called a convective-

diffusive equation as reflected in the −v∂u

∂x(convection with velocity v) and D

∂2u

∂x2

(diffusion with diffusivity D) terms.

In order to have a well-posed (complete) PDE problem specification, auxiliary

conditions must also be specified. For example, in the case of (1.5), which is first-

order in t , and second-order in x, one initial condition (IC) is required (for t ), and

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two boundary conditions (BCs) are required (for x). These might be, for example,

u(x, 0) = f(x) (1.6)

u(0, t ) = u0 (1.7)∂u(L, t)

∂x= 0 . (1.8)

The notation for specifying auxiliary conditions is to denote the specific value of 

the independent variable. For example, t  = 0 is specified in initial condition (1.6) as

u(x, 0) [an alternative would be to write u(x,t  = 0)]. Boundary conditions are of 

three types:

BC Type Example

Dirichlet u(0, t ) = u0(t )

Neumann∂u(L,t)

∂x= ux (L,t) = g(t)

third-type D∂u(0, t )

∂x= Dux (0, t ) =

or Robin v(u(0, t )

−u0(t))

Note that BCs are generally specified at different values of the independent variable,

e.g., x = 0 and x = L, while ICs are specified at a single value of the independent

variable, e.g., t  = 0. As the names imply, boundary conditions typically reflect what

is happening at the boundaries of a physical system, and initial conditions specify

how the system starts out (and then evolves according to the PDE).

 Dirichlet BCs specify the value of the dependent value at a specific value of the

independent variable such as (1.7), while  Neumann BCs specify the derivative of 

the dependent variable with respect to the independent variable such as (1.8). A

combination of Dirichlet and Neumann BCs is termed a boundary condition of thethird type or a Robin BC.

Equation (1.5) is an example of a PDE with constant coefficients (assuming the

velocity v and diffusivity D are constant). PDEs can also have coefficients that are

functions of the independent variables, that is, variable coefficients. For example,

Fourier’s second law in cylindrical coordinates (using the Laplacian in cylindrical

coordinates from Table 1.3)

∂u

∂t  = −v

∂u

∂z +D ∂2u

∂z2 +∂2u

∂r2 +1

r

∂u

∂r +1

r2

∂2u

∂θ 2 . (1.9)

The term1

r

∂u

∂rhas the variable coefficient

1

r(a function of the independent varia-

ble r), and the term1

r2

∂2u

∂θ 2has the variable coefficient

1

r2.

We conclude this discussion of the geometric classification of PDEs by asking

whether this serves a useful purpose. The answer is a threefold “yes.” When we

describe a PDE system as elliptic, hyperbolic, or parabolic, we:

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• Immediately convey a concise description of some of its important mathemat-

ical features. For example, an elliptic problem has no initial value variable.

• Suggest approaches to the numerical solution of the PDE. For example, thesolution of an elliptic problem cannot include initial value integration unless

an initial value variable is added to the elliptic problem, but in such a way that

it will have no final effect on the solution as it evolves numerically.

• Become aware of the potential difficulties in computing a numerical solution.

For example, we can anticipate that a hyperbolic problem might produce dis-

continuities or other difficult mathematical forms that must be accommodated

in the numerical calculation of the solution.

Thus, in using the terminology of geometric classification, we immediately give

a useful description of the PDE problem, and an indication of what must be done to

solve it numerically.

All of the PDEs that have been considered thus far have been linear or first degree.

That is, the dependent variable, u, and all of its derivatives have been to the first power

(the degree should not be confused with the order of the derivative; for example∂2u

∂z2

in (1.9) is first degree, but second order).

A second major classification of PDEs is according to their linearity, that is, linear as just described, or nonlinear, with the dependent variable and/or its derivatives not

first degree (or to the first power). For example,

∂2u

∂z2

3

is second order, but third

degree.

Nonlinear PDEs are an essential part of the PDE mathematical description of many

physical systems. The linearity of a system is an important classification since gener-

ally, mathematical methods for solving nonlinear PDEs are unavailable; that is, gen-

erally we don’t know how to solve nonlinear PDEs mathematically or analytically.

The situation is analogous to that of solving nonlinear algebraic and transcendental

equations; generally this cannot be done mathematically either (there are, of course,

special case exceptions). However, we will observe in subsequent parts of this chap-

ter, and throughout this book, that numerical methods can solve systems of nonlinear

PDEs. In fact, there is no fundamental limit to the solution of nonlinear PDE prob-

lems numerically, although each new problem generally has to be considered on a

case by case basis; that is, numerical procedures have to be developed for the specific

problem system. In fact, the central topic of this book, the adaptive method of lines,

is a general procedure for the numerical solution of nonlinear PDEs.To conclude this preliminary discussion of nonlinearity, consider (1.9) with an

additional term added to the RHS

∂u

∂t = −v

∂u

∂z+ D

∂2u

∂z2+ ∂2u

∂r2+ 1

r

∂u

∂r+ 1

r2

∂2u

∂θ 2

+ k0e−E/(Ru) (1.10)

where k0, E, and R are constants. Note that the exponential e−E/(Ru) contains the

dependent variable, u, in a nonlinear form (this can be confirmed by expanding the

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exponential function in a Taylor series that will include powers of u; therefore in this

series expansion, u is not to the first degree or power, and so (1.10) is a nonlinear

PDE).

Boundary conditions can also be nonlinear. As an example, the boundary condition

k∂u(L,t)

∂x= ε(u4

L − u4(L,t))

is nonlinear because of the term u4(L,t). In general, nonlinear boundary conditions

will preclude an analytical solution to the associated PDE in the same way as if the

nonlinearity appeared in the PDE; in other words, we generally don’t know how to

solve PDEs analytically that have nonlinear boundary conditions.

A third classification of PDEs (in addition to the geometric and linearity classifica-

tions discussed previously) that will have particular relevance in the remainder of this

book is the smoothness of the PDE solutions. Specifically, PDEs can have solutions

that change very abruptly in space and, because of the PDE, they will therefore also

change abruptly in time. Additionally, these abrupt changes, which are also called

steep fronts, can move in space as the solution evolves in time, that is, steep moving

 fronts. In the extreme, the steep moving fronts can be discontinuous (i.e., can be in

the form of discontinuities).

The resolution of steep moving fronts so as to accurately determine where the frontsare, and what form they take, is generally a difficult computational problem in the

numerical solution of a PDE system. We must look at the spatial regions where the

rapid change takes place in greater detail than in the spatial regions where the solution

is relatively smooth. But this implies that we know the location of the rapid changes,

and what form they take, so that we can use enhanced numerical methods in those

regions, even as the location of these regions changes. In other words, the numerical

algorithm for the PDE solution must be adaptive, either through intervention by the

analyst, or automatically as part of the numerical algorithm. This latter characteristic,the adaptive solution of PDEs, is the central topic of this book.

To illustrate how PDEs can propagate mathematical forms that are difficult to

handle numerically, we start with (1.2). If we consider one dimension only, x, and

take the velocity as constant, vx = v, (1.2) becomes the linear advection equation

∂u

∂t = −v

∂u

∂x(1.11)

where u isused inplaceof ρ (the dependent variable of a PDE is commonly designatedas u in the numerical analysis literature).

The simplicity of (1.11) is deceptive because it is also one of the most difficult

PDEs to integrate numerically. To illustrate this, we consider an initial condition

(one is required because of the first-order derivative∂u

∂t )

u(x, 0) = f(x) (1.12)

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and a boundary condition (one is required because of the first-order derivative∂u

∂x)

u(0, t )

=0 . (1.13)

The solution to (1.11), subject to (1.12) and (1.13), is easily derived as

u(x,t) = f (x − vt ) (1.14)

which isknownasa traveling wavesolution since it is the samefunction f  everywhere

in the displaced spatial coordinate x−vt . For example, if v > 0 and the solution starts

out as the initial function of (1.12), the solution will be this same function traveling

left to right with velocity v.

To see how difficult (1.11) can be to solve, we consider as the specific initialcondition function the Heaviside unit step function, h(x)

= 0, x < 0

f(x) = h(x)

= 1, x > 0 .

Thus, from (1.14), we see that the solution to (1.11) for this case is

=0, x

−vt < 0

u(x,t) = h(x − vt )= 1, x − vt > 0 .

This solution has a finite discontinuity (unit jump) at x = vt . In other words, the

solution is a unit step traveling left to right at velocity v. Equation (1.11) with the

discontinuous initial condition h(x) is an example of a Riemann problem.

At x = vt , the derivative∂u

∂xin (1.11) is undefined, so in a sense, this is an

impossible problem to solve numerically. Various approximations for computing

the solution will be considered briefly in the next section as examples of differentapproaches to the Riemann problem.

Finally, if we write the one-dimensional (x only) version of (1.2) as (again with u

in place of ρ)

∂u

∂t = −∂(vu)

∂x(1.15)

or in subscript notation

ut  + F(u)x = 0 (1.16)

where F(u) = vu is a flux function (note again that it has the units of a flux), and

(1.16) is written in conservation form, i.e., it is a conservation law equation. The

Riemann problem for (1.16) has a discontinuous initial condition

= u−, x < 0

u(x, 0)

= u+, x > 0

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where u− and u+ are unequal initial values of u. An extensive numerical analysis

literature exists for thesolution of conservation law equationsandassociated Riemann

problems. In the next section we will consider only a few basic numerical methods

for these problems.

1.2 The Method of Lines

Consider the PDE problem

ut  = f (u), xL < x < xR, t > 0 (1.17)

where

ut  =∂u

∂t u = vector of dependent variables

t  = initial value independent variable

x

=boundary value independent variables

f = spatial differential operator= f (x, t, u, ux, uxx, · · · )

Note that in order to discuss a system of PDEs with a dependent or solution vec-

tor u, a bold-face variable denotes a vector and again, a subscript denotes a partial

derivative. x is a three-vector that, for example, can have components (x,y,z) in

Cartesian coordinates, (r,θ,z) in cylindrical coordinates, and (r,θ,ϕ) in spherical

coordinates. Equation (1.17) is therefore quite general, and can encompass all of the

PDEs considered previously in one, two, and three spatial dimensions plus time. For

example, if f (x, t, u, ux, uxx , · · · ) = −vux , we have the scalar advection equation

(1.11).

The method of lines (MOL) is a computational approach for solving PDE problems

of the form of (1.17) that proceeds in two separate steps: first, spatial derivatives,

e.g., ux, uxx , · · · , are approximated using, for instance, finite difference (FD) or finite

element (FE) techniques. Second, the resulting system of semi-discrete ODEs in the

initial value variable is integrated in time, t .

To illustrate the MOL, we again consider the linear advection equation (1.11)

approximated on a spatial grid in x of N  grid points separated uniformly by a distance

x. If the spatial derivative∂u

∂xis replaced with a second-order, centered FD at grid

point i,

∂u

∂x= −v

ui+1 − ui−1

2x+ O(x2), i = 1, 2, · · · , N  (1.18)

then substitution of this approximation in (1.11) with v = 1 gives a system of N 

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ODEs

dui

dt  = −

ui+1 − ui−1

2x

, i

=1, 2,

· · ·, N  (1.19)

Note spatial grid index i has the values corresponding to a system of N  initial value

ODEs that can be integrated by a library ODE integrator. Of course, in the process,

the initial condition (1.12) must be specified at the N  grid points; also, u0 and uN +1

are fictitious points (outside the spatial domain) that must be included in the ODEs

for i = 1 and i = N  (methods for using boundary conditions to handle boundary and

fictitious points will be considered subsequently).

If the solution to (1.19) is computed for two unit step functions, separated by an

interval in t  of 50 (a square pulse), with N  = 101 (or 100 intervals of length x), theMOL solution is oscillatory as indicated in Figure 1.1(a) (the solid line is the exact

solution). This is an example of the first form of numerical distortion, i.e., numeri-

cal oscillation, and for this example, the oscillation does not diminish significantly

with increasing numbers of grid points, N  (even though the FD approximation is

second order, i.e., O(x2), which indicates that as x decreases, the error of the

FD approximation decreases as x2, depending on some conditions that we will

not discuss here). Thus, we come to the conclusion that even though the individual

terms (derivatives) in a PDE are approximated by what seems to be reasonable (ac-

curate) approximations, when these approximations are substituted in the PDE, theresulting numerical solution can be highly inaccurate. In fact, a large part of what

is discussed subsequently in this book pertains to the choice and implementation of 

approximations that give accurate solutions to the PDEs.

In general, the accuracy of the numerical solution will depend on the smoothness

of the actual (analytical) solution. For the preceding problem, the solution u(x,t) has

twojump discontinuities, andasa consequence, thecentered FDapproximation of ∂u

∂x,

producesunrealisticoscillations. If, however, the initial conditionis notdiscontinuous

(as it was with the square pulse in Figure 1.1(a)), but rather, is a triangular pulse that

is continuous in u(x,t), but discontinuous in∂u

∂x, the MOL solution is much closer

to the true solution as indicated in Figure 1.1(b).

If the initial condition is a smooth cosine pulse, the agreement between the MOL

and the true solution is even better as demonstrated in Figure 1.1(c). Thus, we see that

the performance (accuracy) of a particular approximation of the PDE depends on the

conditions of the problem, in this case, the smoothness of the initial condition. If the

initial condition has a jump or discontinuity [as does u(x, 0) = h(x)], this jump willpropagate in space and time, which is a hallmark characteristic of hyperbolic PDEs.

The preceding example demonstrates the essential features of the MOL solution of 

PDEs, i.e., algebraic approximation of the spatial derivatives, followed by integration

of the resulting system of initial values ODEs. We also observed that the approxi-

mation of the spatial derivatives is a critical step. In the next section, we consider

other approximations to the spatial derivative in (1.11),∂u

∂x, that could conceivably

give better solutions than in Figures 1.1(a), (b), and (c).

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FIGURE 1.1(1.11) approximated as (1.19).

1.2.1 Spatial Discretization

In this book dedicated to PDE problems developing steep spatial moving fronts,

approximation of first-order (convective) terms (i.e., ux) will play a central role. To

avoid undesirable oscillations in the solution profiles, it is generally necessary to re-

sort to upwind spatial approximations. A variety of methods are available, includingupwind finite differences, upwind orthogonal collocation, TVD (total variation di-

minishing) schemes [11] such as flux limiters and ENO (essentially non-oscillatory)

schemes, etc. Recently, TVD centered methods have also been proposed [19], which

have the advantage that no a priori information on the flow direction is required.

To demonstrate the idea of upwinding, we now use as the approximation of ∂u

∂xin

(1.11) the first-order, two point upwind FD

∂u∂x

= ui − ui−1

x+ O(x), i = 1, 2, · · · , N  (1.20)

so that the system of ODEs becomes

dui

dt = −v

ui − ui−1

x, i = 1, 2, · · · , N . (1.21)

As we see in Figure 1.2(a), the numerical oscillation from the centered approxima-

tion considered previously, (1.18), has been eliminated, but now we have excessive

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rounding or numerical diffusion in the MOL solution, which is the second major

form of numerical distortion, i.e., we have now observed numerical oscillation and

numerical diffusion. Further, this numerical diffusion is not substantially reduced by

increasing N , and it also persists for the triangular and cosine pulses, as shown in

Figures 1.2(b) and (c). The approximation for∂u

∂x, (1.20), is called an upwind FD

because it uses, in addition to the point of the approximation, i, the point upwind,

i − 1 (for v > 0), but not the point downwind, i + 1, as in the preceding centered

approximation (1.18) (i + 1 would be the upwind point for v < 0 so that generally

for upwind approximations, we need to know the direction of flow, i.e., the sign of 

the velocity).

 

FIGURE 1.2

(1.11) approximated as (1.21).

At the boundaries corresponding to i = 1 and i = N , we can take as the ODEs

u1

=0,

du1

dt  =0 (1.22)

corresponding to BC (1.13), and

duN 

dt = uN  − uN −1

x(1.23)

so that (1.23) does not involve the fictitious point i = N  + 1.

What has been done with centered and upwind FD approximations, (1.18) and

(1.20), cannot be improved to the point that numerical oscillation and diffusion are

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essentially eliminated in the MOL solution. In other words, linear approximations

of spatial derivatives [note that the dependent variable u appears linearly in (1.18)

and (1.20)] will have these distortions to varying degrees. To essentially eliminate

oscillationand substantially reducediffusion, nonlinearapproximationsmustbeused.A spectrum of nonlinear approximations can be considered, e.g., ENO methods.

Here we briefly mention  flux limiters. In  Figures 1.3(a), (b), and (c), the MOL

solution to (1.11) for the square, triangular, and cosine pulses are given with the

van Leer flux limiter used to approximate∂u

∂x; we observe good agreement with the

analytical solutions. In Figures 1.4(a), (b), and (c), the corresponding MOL solutions

are given for the Smart flux limiter. Thus, we observe that for the problem of (1.11),

nonlinear approximations (such as flux limiters) are effective for computing accurate

numerical solutions, and more generally for hyperbolic problems with steep moving

fronts, these nonlinear approximations give accurate solutions. Thus, these and other

approximations for these difficult problems will be considered subsequently. As the

name “flux limiter” suggests, the value of the flux function F(u) in (1.16) is limited

to avoid numerical distortion in the numerical solutions, [e.g., of (1.11)], particularly

the elimination of numerical oscillation.

1.2.2 Time Integration

Spatial discretization usually produces a system of stiff  ODEs (ODEs with widely

separated eigenvalues). As the stability restriction of an explicit time integration

method is inversely proportional to some power of the grid spacing, x (this power

is usually equal to the order of the highest spatial derivative) [14], the time step

restriction in a finely gridded region (with small x) can be much more severe than

in coarsely gridded regions. Hence, standard explicit Runge–Kutta methods may be

computationally inefficient and implicit methods, e.g., BDF (backward differentiation

formula) or implicit RK solvers, are better suited to solve these problems. The choice

of an ODE integrator is an important aspect of the MOL solution of PDE systems.

However, we will not consider various ODE integration algorithms and the associated

computer codes in detail at this point in order to limit this discussion of the MOL

to a reasonable length. Rather, the choice of an ODE integrator will be addressed

through example applications in the remainder of this chapter and in the subsequent

chapters. Fortunately, a broad choice of quality library ODE integrators is available

and, in fact, one of the major advantages of the MOL approach to PDE systems is the

opportunity to use the advances in ODE integrators and their associated codes.

1.3 Adaptive Grid Methods

In the classical MOL, whereas the time step size is automatically adjusted by

the ODE solver, the spatial grid xi , i = 1, 2, . . . , N  , is usually held constant over

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FIGURE 1.3(1.11) approximated with van Leer flux limiter.

 

FIGURE 1.4

(1.11) approximated with Smart flux limiter.

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the complete integration time interval. However, if for the problem under study,

the model PDEs develop steep moving fronts (such as shock waves in compressible

flows, phase boundaries during nonequilibrium thermal processes, etc.), a very fine

spatial grid is required over the whole spatial domain to capture and resolve the highspatial gradients. Outside of these regions of high spatial activity, a large number

of nodes are “wasted,” resulting in unnecessary computational expense and lack of 

spatial resolution of important small-scale solution features.

Over the last 20 years, a great deal of interest has developed in procedures with

time and space adaptation and various sophisticated techniques have been proposed.

Adaptive grid methods can be classified in several ways, according to the criterion

used to update the grid and to the temporal evolution of the grid point number and

location. Following a set of reference papers [6, 8, 12, 14, 29], these several conceptsare introduced in the next sections.

1.3.1 Grid Adaptation Criteria

One of the major approaches for defining the spatial node movement is based on

the equidistribution principle, i.e., the grid points xi , i = 1, 2, . . . , N  are moved so

that a specified quantity, also called the monitor function m(u), is equally distributed

over the spatial domain, i.e., xi

xi−1

m(u)dx = xi+1

xi

m(u)dx = c, 2 ≤ i ≤ N  − 1 (1.24)

or in discrete form

M i−1xi−1 = M i xi = c, 2 ≤ i ≤ N  − 1 (1.25)

where xi

=xi+1

−xi is the local grid spacing, M i is a discrete approximant of the

monitor function m(u) in the grid interval [xi , xi+1], and c is a constant.Equidistribution principles have been used in many different ways to numerically

solve PDEs having solutions with steep moving fronts. One of the earliest attempts

is due to White [36], who used the arc-length of the solution

m(u) = 

α + ux22

. (1.26)

Another possibility is to equidistribute a measure of the local curvature or spatial

truncation error, e.g., [15]

m(u) = uxx . (1.27)

The second major approach todefine the nodemovement is tominimize a functional

depending on error measures and/or grid structure properties. A method belonging to

this latter category is described by Hyman [14] and Petzold [23] who define the grid

movement by minimizing a measure consisting of a combination of node velocities

and time derivatives of the solution. The minimization of both the time variation in

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the solution and the time variation in the grid leads to a slowly varying Jacobian of 

the semi-discrete ODE system and in turn, to reduced computational expense.

Probably the most important representative in this category is the moving finite

element method of Miller et al. [9, 20, 21], where the error measure is the square of the residual of the PDE written in finite element form. ODEs for the solution and

the nodal (grid point) positions are obtained by minimizing the integral of this error

measure with respect to the time derivatives of the nodal positions and amplitudes.

1.3.2 Static vs. Dynamic Gridding

Following the MOL philosophy, static gridding algorithms proceed in four basic

steps:1. approximation of the spatial derivatives on a fixed nonuniform grid

2. integration of the resulting semi-discrete ODEs over N adapt time steps

3. adaptation of the spatial grid

4. interpolation of the solution to produce initial conditions on the new grid

The main advantageof this approach is that the PDE solutionand the gridadaptation

procedure are uncoupled. Hence, it is easily implemented and allows the use of several artifices such as a variable number of nodes (i.e., grid refinement). The main

disadvantages are: (1) the time integration is halted periodically (or at every time step

if N adapt = 1) to adapt the spatial grid (resulting in a computational overhead due to

the frequent solver restarts); consequently, as the grid points are moving at discrete

times only, they may be ineffectively placed (resulting in large temporal gradients

when a steep moving front crosses some of the grid points, and therefore the ODE

solver is required to use extremely small step sizes to retain accuracy), and (2) an

interpolation technique is required to transfer the data from the old to the new grid.

Another approach is to move the grid points continuously in time, i.e., to use

dynamic gridding, so that their locations follow the moving front and remain near

optimal. This way, the moving front is less likely to cross a grid point and longer time

steps can be taken. The development of this approach requires the introduction of the

 Lagrangian formulation [8] of the PDE problem (1.17). For this purpose, consider

the continuous time trajectories of the grid points

xR = x1 < x2(t) < · · · < xi (t) < · · · < xN  = xR . (1.28)

Along x(t) = xi (t ) the total temporal derivative of u is given by.u= ut +

.x ux = f (u)+ .

x ux . (1.29)

The ODEs defining the grid point movement, i.e.,.

x = g(t), can be derived based

on some physical a priori knowledge, such as a flow-related quantity. For example,

in the case of the advection equation

ut  = −vux (1.30)

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a natural choice is to attach the node movement to the fluid velocity, i.e.,.

x = v, so that.u (x(t),t) = 0 along the characteristic curves. This very simple example highlights

the main objective of this approach, i.e., to minimize the temporal variation of the

solution (in the moving reference frame) so as to allow the largest possible time stepsizes.

Another approach for defining the grid point movement is to express a spatial

equidistribution principle in differential form so as to equally distribute a monitor

functionsuch as the arc length of the solution (1.26). In this case, gridpoint movement

ensures a smoothing of the problem in space, but does not necessarily reduce the

temporal variation of the solution.

As stressed in [8], it is generally not possible to fulfill both objectives, i.e., temporal

and spatial smoothing, simultaneously.

1.3.3 Moving Grid and Grid Refinement Algorithms

Grid refinement algorithms are methods that change the number of nodes as time

evolves. Most of these methods start with a fixed global grid (or base grid) and

proceed to locally add nodes in the regions of highspatial activity and to remove nodes

outside of these regions. Refinement is essential if solutions are to be calculated to a

prescribed level of accuracy (this approach is close to that used in ODE solvers whereas many time steps as needed are taken to bring the local truncation error to within

prespecified tolerances). As a consequence of local node additions and removals, the

overall grid structure may become complicated in the sense of the accompanying data

structures and internal boundary treatment between fine and coarse grids. As the grid

is adapted at discrete time levels only, local refinement methods belong to the static

gridding category.

On the other hand, moving grid methods concentrate a constant number of nodes

in the regions of high spatial activity. Node movement can be accomplished contin-

uously in time, i.e., these methods fall into the dynamic gridding category. However,

with a fixed number of nodes, local resolution is achieved at the expense of depre-

ciating the resolution in other regions. The situation becomes critical when there

are not enough nodes to describe the complete solution profile. For example, in the

case of several steep moving fronts acting in different regions of the spatial domain,

the numerical computation encounters problems if the grid is following one front and

another one arises somewhere else. Since the number of nodes is fixed throughout the

entire course of the computation, no new grid structure is created for the new front,

but rather the old grid has to adjust itself abruptly. This incorrect transient restrictsthe size of the time steps and diminishes the overall efficiency of the method. These

problems can be alleviated by combining node movement and grid refinement.

1.3.4 Grid Regularity

The accuracy of the spatial derivative approximation and the stiffness of the result-

ing system of semi-discrete ODEs is determined to a large extent by the regularity

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of the grid point spacing and the smoothness of the ODE temporal trajectories. As

adaptive grid methods tend to concentrate the grid points in regions of high spatial

activity, spatial grid distortion as well as abrupt change in the node distribution must

be limited through spatial and temporal regularization procedures (for example, usingconstraints on the grid spacing in the spatial equidistribution procedure, penalty terms

in the functional to be minimized, etc.). These regularization procedures always make

the method more complicated in the sense that they involve a set of additional tuning

parameters.

1.4 Case Studies

This section is devoted to the numerical study of several application examples taken

from physics and engineering. At this stage, we would like to illustrate the various

concepts introduced thus far, to show the diversity of adaptive grid algorithms and

to demonstrate their potential to address challenging applications. Of course, the

selection of particular methods and applications is intended only to illustrate basic

concepts and methods, and this section is by no means exhaustive. What follows is

 just a sampling of the spectrum of adaptive grid methods published in the literature

over the last 20 years.

1.4.1 Case Study 1

As a first test-example, consider Burgers’ equation

ut  = −u ux + uxx , 0 < x < 1, t > 0 (1.31)

with initial and Dirichlet boundary conditions taken from the exact solution

u(x,t) = (0.1r1 + 0.5r2 + r3)/(r1 + r2 + r3) (1.32)

where r1(x,t) = e(−x+0.5−4.95t)/20ε, r2(x,t) = e(−x+0.5−0.75t)/4ε, and r3(x,t) =e(−x+0.375)/2ε. By adjusting the numerical value of the viscosity coefficient , a broad

spectrum of convection-diffusion problems can be generated. Here, we consider a

medium value of  = 0.001, which leads to moderate front steepening.

This problem is solved using ANUGB, a local refinement algorithm developed by

Hu and Schiesser [15]. In this method, the grid adaptation criterion is based on the

solution curvature. A uniform base grid xi , i = 1, . . . , N  b, which is the foundation

upon which all subsequent grids are built, is first defined. The second-order derivative

of the solution is estimated at each of the base grid points in order to locate the regions

of high spatial activity; a set of threshold values for uxx (xi ) controls the addition

or subtraction of new grid points. When uxx (xi ) exceeds one of the user-specified

levels, the algorithm inserts a certain number of grid points in the base grid intervals

[xi−1, xi] and [xi , xi+1] (the number of new grid points depending on the magnitude

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of uxx (xi )). As another tuning parameter, the influence of the base grid point xi

can be extended beyond xi−1 and/or xi+1 by specifying the sizes of buffer zones on

both sides of the base grid point. As the grid is updated at discrete time levels only,

a buffer zone in the flow direction allows a correct description of the moving frontduring a certain time interval (i.e., a finely gridded buffer zone compensates for the

inaccurate determination of the front location).

The PDEs are discretized using cubic spline differentiators and the resulting system

of semi-discrete equations is integrated in time using the implicit RK solver RADAU5

[10] with error tolerances set to atol = rtol = 10−5. The time integration is halted

every N adapt = 5 integration steps in order to insert or remove grid points. The initial

values on these new grid points are interpolated using cubic splines (to provide initial

conditions for the new ODEs). Numerical results are depicted in Figure 1.5 whichshows the evolution of the spatial profile at t  = 0, 0.2, 0.4, . . . , 1. The bottom of the

figure indicates the location of the grid points, i.e., the fixed N b = 51 base grid points

and the finer grids following the moving front.

FIGURE 1.5

Burgers’ equation: adaptive grid solution using ANUGB on a base grid with 51

nodes (dots) and exact solution (solid line) every 0.2 units in time.

This method is very intuitive and works quite well. However, one of the main

drawbacks is that a relatively large number of base grid points is needed to sense the

solution curvature. Also, in contrast with the apparent simplicity of the algorithm, the

user has the requirement to adjust several parameters, e.g., threshold values, number

of points in the finer grids, extension of the buffer zones, so that tuning might become

quite involved. Additional applications of this method are presented by the authors

in [31].

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1.4.2 Case Study 2

Consider a bio-engineering application [28], e.g., a fixed-bed bioreactor in which

biomass growth and death processes take place

ν1S ϕ1→ X + ν2P  (1.33)

Xϕ2→ Xd  (1.34)

The biomass (micro-organisms) X grows on the fixed bed due to the substrate S 

dissolved in the flowing medium. In addition to biomass, a product of interest P  is

also obtained. Simultaneously, a part Xd  of the biomass dies.

It is assumed that the substrate diffusion is negligible, so that the following mass

balance PDEs can be written

S t  = −vS x − ν11 −

ϕ1, 0 < x < L, t > 0 (1.35)

Xt  = ϕ1 − ϕ2 (1.36)

where the same notation is used for the component concentrations.

In (1.35), v = F/(A) is the superficial velocity of the fluid flowing through the

bed, is the total void fraction (or bed porosity), and ϕ1 is the growth rate given by

a model of Contois

ϕ1 = µmaxS 

kcX + S X (1.37)

where µmax is the maximum specific growth rate and kc is the saturation coefficient.

In (1.34), ϕ2 is the death rate given by a simple linear law

ϕ2 = kd X . (1.38)

The model PDEs (1.35) and (1.36) are supplemented by a Dirichlet boundary condi-

tion in x = 0 and initial conditions.

The numerical values of the model parameters are: L = 1 m, A = 0.04 m2,

= 0.5, F  = 2 l/ h, ν1 = 0.4, µmax = 0.35 h−1, kc = 0.4, kd  = 0.05 h−1.

This application is analyzed with the local refinement code PARAB, which has

been developed by the authors and implements a method originally proposed by

Eigenberger and Butt [5]. The grid placement criterion is based on an error estimation

procedure, i.e., the solution is represented by piecewise second-order parabolas and

the interpolation error between two subsequent parabolas (defined on xi−2, xi−1, xi

and xi−1, xi , xi+1) in the middleof each grid interval is evaluated. If this error is above

a specified maximal error level Emax, an additional node is inserted. Conversely, if 

two subsequent errors are below another specified minimal error level Emin, the node

in between is removed. This basic algorithm has been modified so that, if the solution

displays a relatively flat profile at some time, not all the nodes are removed from

the grid (here, there is no fixed base grid so that all the nodes could be deleted one

after another if the interpolation error is small) and the method is able to cope with

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the sudden appearance of new steep profiles, i.e., a maximal grid spacing xmax is

specified.

Spatial approximation is accomplished using 5-point biased upwind finite differ-

ences. The tuning parameters of PARAB are selected as follows: Emin = 10−5,Emax = 10−4, xmax = 0.2. Error tolerances atol = rtol = 10−4 are imposed for

the time integration with RADAU5. The solver is halted every N adapt = 3 integra-

tion steps for updating the grid. Data are transferred from the old to the new grid

using cubic spline interpolation. During the course of the computation, the number

of grid points varies between 223 and 59. The grid refinement algorithm is easy to

tune and performs quite well. Figures 1.6 and 1.7 show the evolution of the substrate

and biomass concentrations every 5000 sec following a step-change in the biomass

concentration at the reactor inlet from 5 g/ l to 8 g/ l. A steep front of substrateconcentration propagates towards the bioreactor outlet.

1.4.3 Case Study 3

We now turn to a prototype model for oil reservoir simulation taken from [19], i.e.,

the convection-diffusion Buckley-Leverett equation

ut  + f (u)x = ε(ν(u)ux )x , 0 < x < 1, t > 0 . (1.39)

In this expression, the diffusion coefficient ν(u) vanishes for u = 0 and 1,

ν(u) = 4u(1 − u) (1.40)

so that (1.39) is a degenerate parabolic equation. The flux function including gravi-

tational effects is given by

f(u) = u2

u2 + (1 − u)2

1 − 5(1 − u)2

. (1.41)

The PDE (1.39) is supplemented by Riemann initial conditions

u(x, 0) = 0, 0 ≤ x ≤ 1 − 1/√ 

2

= 1, 1 − 1/√ 

2 ≤ x ≤ 1 (1.42)

and Dirichlet boundary conditions in xL = 0 and xR = 1

u(0, t ) = 0

u(1, t ) = 1 . (1.43)

This difficult problem is solved for ε

=0.01 using the moving grid code AGE

[26], which is based on a method published by Sanz-Serna and Christie [25] and anextension proposed by Revilla [24].

AGE is a static gridding algorithm that equidistributes a functional m(u) based on

the solution curvature, i.e., xi

xi−1

m(u)dx = xi+1

xi

m(u)dx, 2 ≤ i ≤ N  − 1 (1.44)

m(u) = 

(α + uxx ∞) . (1.45)

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FIGURE 1.6

Substrate concentration in a fixed-bed bioreactor; adaptive grid solution usingPARAB every 5000 units in time.

FIGURE 1.7

Biomass concentration in a fixed-bed bioreactor; adaptive grid solution using

PARAB every 5000 units in time.

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The scaling factor α can be used to modify the relative importance of values of 

x and values of u. An additional parameter β is introduced to avoid the excessive

clustering of nodes in regions where

uxx

∞ is large, i.e., the values of the second-

order derivative that exceed β are reduced to the value β.A superbee flux limiter is used for computing f(u)x , whereas a 5-point centered

finite difference scheme is applied to the diffusion term. The semi-discrete ODEs are

solved using RADAU5 with error tolerances set to atol = rtol = 10−4. Time integra-

tion and grid adaptation proceed alternately (N adapt = 1). The tuning parameters of 

the adaptive grid algorithm are set as: N  = 101, α = 10−3, and β = 103. Excellent

numerical solutions are obtained, which are graphed in Figure 1.8 (also the figure on

the book cover) every 0.4 units in time. Additional applications of this method are

reported by the authors in [26, 27].

FIGURE 1.8

Buckley-Leverett equation: numerical solution on an adaptive grid with N  =101 nodes (AGE) at t  = 0, 0.4, . . . , 2.8.

1.4.4 Case Study 4

Consider a laboratory-scale catalytic fixed-bed reactor in which the hydrogenation

of small amounts of CO2 to methane is accomplished

CO2 + 4H 2 → CH 4 + 2H 2O . (1.46)

Based on experimental studies, a model of the transient behavior of this plant has

been derived in [30], and investigated numerically in [5]. The model consists of two

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PDEs for the mass balance of CO2 and the energy balance, respectively,

ct 

= −vcx

+Dcxx

−r, 0 < x < L, t > 0 (1.47)

T t  = −v ρgcpg

ρcp

T x + λ

ρcp

T xx + 2kw

rρcp

(T w − T ) + (−H )

ρcp

r (1.48)

with the reaction rate

r = kr

ce−E/RT 

1 + kcc(1.49)

and the boundary and initial conditions

cx (0, t ) = v

D(c − cin ), T x (0, t) = vρgcpg

λ(T  − T in ) (1.50)

cx (L,t) = 0, T x (L,t) = 0 (1.51)

c(x, 0) = c0(x), T (x, 0) = T 0(x) . (1.52)

The numerical values of the model parameters are: L = 0.2 m, r = 0.01 m,

= 0.6, v = 1.5 m/s, D = 5 × 10−4 m2/s, ρcp = 364 kcal/m3K , cpg =2.29 kcal/kgK , ρg = 0.0775 kg/m

3, λ = 3.5 × 10−

4kcal/msK , kw = 5 ×

10−4 kcal/m2sK , T w = 300 K , −H  = 6.006 kcal, kr = 0.971 × 1013m−3s−1,

kc = 12.7, E = 25.211 kcal/mole, R = 1.98 cal/mole − K .

The concentration and temperature transients at reactor start-up, e.g., due to step

changes in the feed concentration cin(t ) = 0 → 2.5 mole% and temperature

T in (t ) = 300 → 500 K , are studied numerically using the dynamic gridding soft-

ware package MOVGRD (ACM 731) [1, 33]. MOVGRD is based on a nonlin-

ear Galerkin discretization of the Lagrangian description of the PDEs (1.29) and

a smoothed equidistribution principle using regularization techniques reported by

Dorfy and Drury [3].

To introduce this method, consider the spatial equidistribution equation (1.25)

expressed in terms of the grid density ni = 1/xi

ni−1

M i−1= ni

M i, 2 ≤ i ≤ N  − 1 (1.53)

where M i is a discrete approximation of an arc-length monitor function (1.26).

In order to avoid excessive spatial distortion and temporal oscillation of the grid,two regularization procedures are used.

First, spatial smoothing is accomplished by replacing the grid density ni in (1.53)

by

n0 = n0 − κ(κ + 1)(n1 − n0)

ni = ni − κ(κ + 1)(ni+1 − 2ni + ni−1) 2 ≤ i ≤ N  − 1 (1.54)

nN  = nN  − κ(κ + 1)(nN −1 − nN )

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where κ is a positive parameter. The introduction of the “anti-diffused” density ni

ensures that the grid is locally bounded, i.e., that adjacent grid spacings do not differ

substantially from one another (the complete developments can be found in [33])

κ

κ + 1≤ ni−1

ni

≤ κ + 1

κ (1.55)

Second, temporal smoothing is accomplished by replacing the system of algebraic

equations (1.53) by a system of differential equations

ni−1 + τ .

ni−1

M i−1 =

ni + τ .

ni

M i

, 2

≤i

≤N 

−1 (1.56)

where the positive parameter τ  acts as a time-constant preventing abrupt changes in

the grid movement. Experience shows that spatial smoothing is more important than

temporal smoothing. The semi-discrete approximation of the Lagrangian form of the

PDEs is combined with Equation (1.56) to yield a system of ODEs that is integrated

using the BDF solver DASSL [22].

Figures 1.9 and  1.10 show the evolution of the concentration and temperature

profiles every 100 s. Reaction takes place in the middle of the tubular reactor, resulting

in the formation of a “hot spot.” These numerical results have been obtained withthe parameter values: N  = 51, α = 10−4, κ = 2, and τ  = 10−3. Tolerances

atol = rtol = 10−7 are imposed for the time integration with DASSL.

FIGURE 1.9

Fixed-bed methanator: concentration profiles on an adaptive grid with N  = 51

nodes (MOVGRD) at t  = 0, 100, . . . , 1000.

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FIGURE 1.10

Fixed-bed methanator: temperature profiles on an adaptive grid with N  = 51

nodes (MOVGRD) at t  = 0, 100, . . . , 1000.

1.4.5 Case Study 5

Consider a model of flame propagation [4] consisting of two coupled equations for

mass density and temperature

ρt  = ρxx − N DAρ

T t  = T xx + N DAρ, 0 < x < 1, t > 0 (1.57)

where N DA = 3.52 × 106 e−4/T .

The initial conditions are given by

ρ(x, 0) = 1, T (x, 0) = 0.2, 0 ≤ x ≤ 1 (1.58)

and the boundary conditions are

ρx (0, t ) = 0, T x (0, t ) = 0 ,

ρx

(1, t )=

0, T (1, t )=

f (t ), t  ≥

0 , (1.59)

with

f (t ) = 0.2 + t/2 × 10−4, t  ≤ 2 × 10−4

= 1.2 t  ≥ 2 × 10−4 . (1.60)

The heat source located at x = 1 generates a flame front that propagates from right

to left at an almost constant speed.

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Thisproblem issolvedonthe timeinterval (0,0.006) using themoving grid software

MOVCOL [17]. In MOVCOL, the node movement is governed by a continuous

moving mesh equation (or moving mesh PDE - MMPDE) instead of a discrete moving

mesh equation as in MOVGRD [i.e., the system of ODEs (1.56)]. MMPDEs can bederived in several ways, as reviewed in [16]. Here, we briefly sketch the derivation

steps of a basic MMPDE based on the equidistribution principle.

A one-to-one transformation between physical (x) and computational (ξ ) coordi-

nates is introduced

x = x(ξ , t ), ξ  ∈ [0, 1]x(0, t ) = xL, x(1, t ) = xR (1.61)

by equidistributing a monitor function m(x,t), i.e., x(ξ,t)

xL

m(z,t)dz = ξ 

 xR

xL

m(z,t)dz . (1.62)

Differentiating (1.62) with respect to ξ  once and twice yields two differential forms

of the equidistribution principle

m(x(ξ,t),t)∂x(ξ,t)

∂ξ  =  xR

xL

m(z, t) dz . (1.63)

∂

∂ξ 

m(x(ξ,t),t)

∂x(ξ,t)

∂ξ 

= 0 (1.64)

which are called quasi-static equidistribution principles since they do not contain the

node speed.

x (ξ, t ).

To derive an MMPDE, we require that the mesh satisfies the latter equation at the

time t 

+τ  instead of at t , so that

∂

∂ξ 

m(x(ξ, t + τ ) , t  + τ )

∂x(ξ,t  + τ )

∂ξ 

= 0 (1.65)

gives a relaxation time τ  for the mesh to satisfy the equidistribution principle.

Using the Taylor series expansions

m(x(ξ, t + τ ) , t  + τ ) = m(x(ξ,t),t) + τ .

x (ξ, t )∂m(x(ξ,t),t)

∂x

+ τ 

∂m(x(ξ,t),t)

∂t  + O(τ 2

)

∂x(ξ,t  + τ )

∂ξ = ∂x(ξ,t)

∂ξ + τ 

∂.

x (ξ, t )

∂ξ + O(τ 2) (1.66)

an MMPDE is obtained as

∂

∂ξ 

m

∂.

x

∂ξ 

+ ∂

∂ξ 

∂m

∂ξ 

.x

= − ∂

∂ξ 

∂m

∂t 

∂x

∂ξ 

− 1

τ 

∂

∂ξ 

m

∂x

∂ξ 

(1.67)

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that can be further simplified by dropping ∂∂ξ 

∂m∂t 

∂x∂ξ 

as well as ∂

∂ξ 

∂m∂ξ 

.x

, i.e.,

∂∂ξ 

m ∂

.

x∂ξ 

= − 1τ 

∂∂ξ 

m ∂x

∂ξ 

. (1.68)

This simplification is justified by the fact that the term involving ∂m∂t 

is often difficult

to compute and is not absolutely necessary since the term − 1τ 

∂∂ξ 

m ∂x

∂ξ 

is a source of 

node movement which measures how closely the mesh satisfies the equidistribution

principle, even when m(x,t) is independent of t .

Of course, temporal mesh smoothing is automatically built into the MMPDE. How-ever, for PDE problems involving large solution variations, the monitor function

m(x,t) is generally fairly nonsmooth in space, and spatial smoothing must be intro-

duced [18]. To this end, a potential approach is to replace the monitor function m by

a “smoothed” M , which satisfies a PDE in ξ  and t  involving an artificial diffusion

term

M 

−

1

λ2

∂2M 

∂ξ 2

=m (1.69)

with boundary conditions

∂M 

∂ξ (0, t) = ∂M 

∂ξ (1, t ) = 0 . (1.70)

However, this approach is not suitable from a computational point of view and

alternatives are developed in [18]. Here, we just mention that MOVCOL is based onthe following smoothed MMPDE:

∂

∂ξ 

1

m

1 − 1

λ2

∂2

∂ξ 2

τ 

.n +n

= 0 (1.71)

for the mesh concentration function n = 1/ (∂x/∂ξ ). The diffusion parameter λ is

of the form (N  − 1) /√ 

γ  (γ  + 1), where γ  is the user defined, spatial smoothing

parameter.MOVCOL uses cubic Hermite collocation for discretization of the physical PDEs

in divergence form, and a three-point finite difference discretization of the MMPDE.

The resulting semi-discrete ODE system is solved using the time integrator DASSL

[22].

The evolution of the temperature and density profiles are graphed every 0.0006 in

Figures 1.11 and 1.12. These numerical results have been obtained with N  = 21,

γ  = 1, τ  = 10−4, and atol = rtol = 10−4.

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FIGURE 1.11

Flamepropagation: temperature profiles on an adaptive grid with N  = 21 nodes(MOVCOL) at t  = 0,0.0006, . . . , 0.006.

FIGURE 1.12

Flame propagation: density profiles on an adaptive grid with N  = 21 nodes

(MOVCOL) at t  = 0,0.0006, . . . , 0.006.

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1.4.6 Case Study 6

Consider a problem taken from [9] describing two countercurrent reactive square

wavesvt  = −0.5vx − vw

wt  = 0.5wx − vw . (1.72)

The initial conditions are

v(x, 0) = 1, 0 < x < 20

= 0, otherwise

w(x, 0) = 1, 80 < x < 100= 0, otherwise . (1.73)

The boundary conditions are

v(0, t ) = v(100, t ) = 0

w(0, t ) = w(100, t ) = 0 . (1.74)

This problem is solved on the time interval (0, 140) using the moving finite element

(MFE) method proposed by Miller and co-workers [9, 20, 21]. In this method, thesolution to (1.17) is approximated using a finite element formulation with piecewise

linear basis functions αj 

u(x,t) ≈ U(x,t) =N 

j =1

U j (t) αj (x, X(t)) , (1.75)

in which both the nodal amplitudes U j (t), j  = 1, . . . , N  , and the nodal positions

xL = X1(t) < X2(t) < · · · < XN (t ) = xR are unknown functions of time.Partial differentiation of (1.75) with respect to time yields

∂U(x,y)/∂t =N 

j =1

U j (t) αj (x,X(t)) + Xj (t) βj (x, X(t)) , (1.76)

in which βj  = −(∂U/∂x) αj  can be considered as a second type of basis function.

The 2N  unknown functions U j (t ) and Xj (t ) are determined by minimizing the L2

norm of the PDE residual

R(U)

22

= ∂U/∂t 

−L(U)

22 with respect to

˙U j (t ) and

Xj (t ), which results in a system of 2N  ODEs

N j =1

αi , αj 

U j  +

αi , βj 

Xj  = αi ,L(U) , (1.77)

N j =1

βi , αj 

U j  +

βi , βj 

Xj  = βi ,L(U) , i = 1, . . . , N . (1.78)

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By reordering the unknown variables in a column vector Y  = (U 1, X1, U 2, X2,

. . . , U  N , XN )T  , it is possible to write (1.77) and (1.78) in a compact form

A(Y)Y  = g(Y ) , (1.79)

where A(Y) is an N  × N  block-tridiagonal matrix (each block is a 2 × 2 matrix

consisting of inner products of the basis functions αj   and βj  ).

Integrating (1.79) in time can become problematic for two reasons. First, the mass

matrix A(Y) becomes ill-conditioned when some nodes drift very close together and

extremely nonuniform grids are generated. Second, the mass matrix A(Y) becomes

singular when parallelism occurs, i.e., when at a particular node Xj   the solution

curvature vanishes. In this case, the MFE method intrinsically fails to determine the

direction in which the node Xj   should be moved.To avoid these problem degeneracies, Miller [20, 21] introduced regularization

terms in the residual minimization, which penalize the relative motions between

nodes. The new minimization problem can be written as follows:

U j , Xj  min ∂U/∂t − L(U)22 +

N j =2

εj Xj  − S j 

. (1.80)

While the ODEs (1.77) remain unchanged, the ODEs (1.78), which govern the nodemotion, become

N j =1

βi , αj 

U j  +

βi , βj 

Xj  + ε2

i Xi − ε2i+1Xi+1

= βi ,L(U) + εi S i − εi+1S i+1 . (1.81)

The left-hand side terms, which involve the internodal viscosities εj  , regularize

the dynamic internodal node movements and keep the resulting mass-matrix positive

definite, while the right-hand side terms, which contain the internodal spring functions

S j  , allow a regularization of the long-term or equilibrium system.

According to a simplified algorithm formulation given in [9], the regularizing

functions are given by

S j  =c1

Xj  − δ, (1.82)

εj  =c2

Xj  − δ , (1.83)

in which c1, c2, and δ are tuning parameters. In particular, δ can be interpreted as a

minimum permissible internodal separation. Both the viscosities and the internodal

spring functions become infinite as the internodal separation approaches δ.

The evolution of the two square waves at t  = 0, 80, 140 is graphed in Figures 1.13,

1.14, and  1.15, respectively. These results have been obtained with 32 elements

(N  = 33 nodes) and the tuning parameters c1 = 10−4, c2 = 10−4, and δ = 10−4.

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FIGURE 1.13

Two countercurrent reactive squarewaves: initial condition(MFE with N  = 29).

FIGURE 1.14

Two countercurrent reactive square waves: interaction at t  = 80 (MFE with

N  = 29).

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FIGURE 1.15

Two countercurrent reactive square waves: propagation at t  = 140 (MFE with

N  = 29).

Time integration is performed with the BDF solver LSODI [13] with error tolerances

atol = 10−2 and rtol = 10−4.

MFE hasattracted considerable attention and, over theyears, several improvements

havebeen proposed, includingmatrixpreconditioning[34, 35]andgradient weighting

[2]. A review of some of these results and additional applications of the method are

presented in [32].

1.5 Summary

We have briefly reviewed the basic computational methods for adaptive MOL and

illustrated their use through a series of example applications. This survey illustrates

the effectiveness of adaptive methods in resolving the sharp spatial and temporal

features of PDE solutions that would be difficult to resolve with fixed grid methods.This discussion is also intended to highlight the computer codes that are readily

available for adaptive methods.

However, the adaptive approach generally involves additional complexity com-

pared to a fixed-grid formulation, including the tuning of method parameters to

achieve the desired solution resolution and accuracy. Thus, some trial and error

is inevitable in using adaptive MOL, and the expectation is that this additional effort

is worthwhile; experience has demonstrated that generally this is the case.

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Now we proceed in the remainder of this book to discussions by developers of 

adaptive methods who elucidate the features of a spectrum of methods. The intention

is to facilitate the use of adaptive methods by highlighting the features of the methods

and associated codes, and by demonstrating the characteristics and effectiveness of the adaptive approach to PDE solutions through example applications.

References

[1] J.G. Blom and P.A. Zegeling, Algorithm 731: a moving-grid interface for sys-

tems of one-dimensional time-dependent partial differential equations, ACM 

Trans. Math. Software, 20, (1994), 194–214.

[2] N. Carlson and K.Miller, Design and application ofa gradient-weighted moving

finite element code, Part I, in 1-D, SIAM J. Sci. Comput., 19, (1998), 728–765.

[3] E.A. Dorfi and L.O’C. Drury, Simple adaptive grids for 1-D initial value prob-

lems, J. Comp. Phys., 69, (1987), 175–195.

[4] H.A. Dwyer and B.R. Sanders, Numerical modeling of unsteady flame propa-

gation, Sandia National Lab. Livermore Report SAND77-8275, 1978.

[5] G. Eigenberger and J.B. Butt, A modified Crank-Nicolson technique with non-

equidistant space steps, Chem. Eng. Sci., 31, (1976), 681–691.

[6] P.R. Eiseman, Adaptive grid generation, Comput. Meth. Appl. Mech. Eng., 64,

(1987), 321–376.

[7] B. Fornberg, Generation of finite difference formulas on arbitrarily spaced grid,

 Math. Comp., 51, (1988), 699–706.

[8] R.M. Furzeland, J.G. Verwer, and P.A. Zegeling, A numerical study of three-

moving grid methods for one-dimensional partial differential equations which

are based on the method of lines, J. Comp. Phys., 89, (1990), 349–388.

[9] R. Gelinas, S. Doss, and K. Miller, The moving finite element method: appli-cations to general equations with multiple large gradients, J. Comp. Phys., 40,

(1981), 202–249.

[10] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II - Stiff and 

 Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991.

[11] A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comp.

Phys., 49, (1983), 357–393.

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[12] D.F. Hawken, J.J. Gottlieb, and J.S. Hansen, Review of some adaptive node-

movement techniques in finite element and finite difference solutions of partial

differential equations, J. Comp. Phys., 95, (1991), 254–302.

[13] A.C. Hindmarsh, ODEPACK: A systematized collection of ODE solvers, in

R.S. Stepleman, ed., Scientific Computing, IMACS, North Holland, 1983, 55–

64.

[14] J.M. Hyman, Adaptive moving mesh methods for partial differential equations,

in Advances in Reactor Computations, American Nuclear Society Press, La

Grange Park, IL, 1983, 24–43.

[15] S.S. Hu and W.E. Schiesser, An adaptive grid method in the numerical method

of lines, in R. Vichnevetsky and R.S. Stepleman, eds., Advances in Computer  Methods for Partial Differential Equations, IMACS, North Holland (1981),

305–311.

[16] W. Huang, Y. Ren, and R.D. Russell, Moving mesh partial differential equations

(MMPDEs) based on the equidistribution principle, SIAM J. Numer. Anal., 31,

(1994), 709–730.

[17] W. Huang and R.D. Russell, A moving collocation method for solving time

dependent partial differential equations, Appl. Num. Math., 20, (1996), 101.

[18] W. Huang and R.D. Russell, Analysis of moving mesh PDEs with spatial

smoothing, SIAM J. Numer. Anal., 34, (1997), 1106.

[19] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear

conservation laws and convection-diffusion equations, UCLA Computational

and Applied Mathematics Report, April 1999.

[20] K. Miller and R. Miller, Moving finite elements, Part I, SIAM J. Numer., 18,

(1981), 1019–1032.

[21] K. Miller, Moving finite elements, Part II, SIAM J. Numer., 18, (1981), 1033–

1057.

[22] L.R. Petzold, A description of DASSL: a differential/algebraic system solver,

in R.S. Stepleman, ed., Scientific Computing, IMACS, North-Holland (1983),

65–68.

[23] L.R. Petzold, Observations on an adaptive moving grid method for one-

dimensional systems of partial differential equations, Appl. Numer. Math., 3,(1987), 347–360.

[24] M.A. Revilla, Simple time and space adaptation in one-dimensional evolu-

tionary partial differential equation, Int. J. Numer. Methods Eng., 23, (1986),

2263–2275.

[25] J.M. Sanz-Serna and I. Christie, A simple adaptive technique for nonlinearwave

problems, J. Comp. Phys., 67, (1986), 348–360.

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[26] P. Saucez, A. Vande Wouwer, and W.E. Schiesser, Some observations on a static

spatial remeshing method based on equidistribution principles, J. Comp. Phys.,

128, (1996), 274–288.

[27] P. Saucez, A. Vande Wouwer, and W.E. Schiesser, An adaptive method of lines

solution of the Korteweg-de Vries equation, Comp. Math. Applic., 35, (1998),

13–25.

[28] N. Tali-Maamar, T. Damak, J.P. Babary, and M.T. Nihtilä, Application of a

collocation method for simulation of distributed parameter bioreactors, Math.

Comp. Sim., 35, (1993), 303–319.

[29] J.F. Thompson, A survey of dynamically-adaptive grids in the numerical solu-

tion of partial differential equations, Appl. Numer. Math., 1, (1985), 3–27.

[30] H. Van Doesburg and W.A. De Jong, Dynamic behavior of an adiabatic fixed-

bed methanator, Int. Symp. Chem. React. Eng., Evanston, Advances in Chem.

Series, 133, (1974), 489–503.

[31] A. Vande Wouwer, P. Saucez, and W.E. Schiesser, Some user-oriented compar-

isons of adaptive grid methods for partial differential equations in one space

dimension, Appl. Numer. Math., 26, (1998), 49–62.

[32] A. Vande Wouwer, P. Saucez, and W.E. Schiesser, Numerical experiments withthe (gradient-weighted) finite element method, submitted.

[33] J.G. Verwer, J.G. Blom, R.M. Furzeland, and P.A. Zegeling, A moving-grid

methodfor one-dimensional PDEsbased on the methodof lines, in J.E.Flaherty,

P.J. Paslow, M.S. Shephard, and J.D. Vasilakis, eds., Adaptive Methods for 

Partial Differential Equations, SIAM, Philadelphia (1989), 160–175.

[34] A.J. Wathen and M.J. Baines, On the structure of the moving finite-element

equations, IMA J. Numer. Anal., 5, (1985), 161–182.[35] A.J. Wathen, Mesh-independent spectra in the moving finite element equations,

SIAM J. Numer., 23, (1986), 797–814.

[36] A.B. White, On the numerical solution of initial/boundary-value problems in

one-space dimension, SIAM J. Numer. Anal., 19, (1982), 683–697.

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Chapter 2

 Application of the Adaptive Method of Lines to Nonlinear Wave Propagation Problems

Alain Vande Wouwer, Philippe Saucez, and William Schiesser

2.1 Introduction

In recent years, much interest has developed in the numerical treatment of PDEs

giving rise to nonlinear wave phenomena, and particularly, solitary waves. In this

chapter, attention is focused on a few particular cases, i.e., the cubic Schrödinger

equation (CSE) and the derivative nonlinear Schrödinger equation (DNLS), as wellas several Korteweg-de Vries (KdV)-like equations in one space dimension. These

equations have been used extensively to model nonlinear dispersive waves in a wide

range of application areas, such as water wave models, laser optics, and plasma

physics.

Inorder toefficientlycompute numericalsolutionsof theseequations, it is appealing

to resort to an adaptive grid technique that automatically concentrates the spatial

nodes in the regions of rapid solution variations (i.e., the wave moving fronts). In

this connection, an adaptive grid refinement algorithm based on the equidistribution

principle and spatial regularization procedures is implemented and applied to severalillustrative problems, including the propagation of a single soliton, the interaction

between two solitons, and the bound state of several solitons, and the propagation of 

a compacton (a soliton with compact support).

Some implementation details are given and the performance of the method is dis-

cussed in terms of accuracy and computational efficiency.

2.2 Adaptive Grid Refinement

In this section, an adaptive grid method which equidistributes a given monitor

function subject to constraints on the grid regularity is presented. The time-stepping

procedure as well as some implementation issues are discussed.

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2.2.1 Grid Equidistribution with Constraints

As we have seen in Chapter 1, spatial equidistribution is an important principle

on which many adaptive grid algorithms are built. If m(u) is a specified monitor

function, the spatial equidistribution equation for the grid points xi , i = 1, 2, . . . , N  ,can be expressed in continuous form xi

xi−1

m(u) dx = xi+1

xi

m(u) dx = c, 2 ≤ i ≤ N  − 1 (2.1)

or in discrete form

M i−1xi−1 = M i xi = c, 2 ≤ i ≤ N  − 1 (2.2)

where xi = xi+1 − xi is the local grid spacing, M i is a discrete approximant of themonitor function m(u) in the grid interval [xi , xi+1], and c is a constant.

A popular monitor function is based on the arc length of the solution [21], i.e.,

m(u) = 

(α + ux22) . (2.3)

Other monitor functions can be used as well and our experience [22] suggests the

use of the following curvature-related function:

m(u)

=  (α

+ uxx

∞) . (2.4)

In this expression, α > 0 ensures that the monitor function is strictly positive and

acts as a regularization parameter which forces the existence of at least a few nodes

in flat (2.3) or linear (2.4) parts of the solution. Another regularization parameter

β > 0 can be introduced [21] to avoid excessive clustering of nodes in regions where

the solution exhibits very steep slope (2.3) or very high curvature (2.4), i.e., β is

used instead of ux22 or uxx ∞ in the evaluation of the corresponding monitor

function. In our experience, this latter regularization is particularly useful in the case

of a curvature-based monitor function (2.4).

The accuracy of the spatial derivative approximations (e.g., using finite differencetechniques) and the stiffness of the semi-discrete system of differential equations

are largely influenced by the regularity and spacing of the grid points. This stresses

the importance of limiting grid distortion using spatial regularization procedures. In

practice, the use of parameters α and β is not sufficient to ensure, in a systematic way,

the grid regularity, and in the following, a more advanced procedure due to Kautsky

and Nichols [12] based on the concept of  locally bounded grid  is explored. This

procedure, as we shall see, involves a variable number of nodes.

A grid is said to be locally bounded with respect to a constraint K

≥1 if 

1

K≤ xi

xi−1≤ K, 2 ≤ i ≤ N  − 1 . (2.5)

Then the equidistribution problem becomes the following.

Given a monitor function m ∈ C+ (the set of continuous piecewise functions on

[xL, xR]) and constants c > 0 and K ≥ 1, find the grid that is

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1. sub-equidistributing with respect to m and c on [xL, xR], i.e., for the smallest

number of nodes N  such that N c ≥ xR

xLmdx, we have

 xi+1

ximdx ≤ c

2. locally bounded with respect to K

The idea of the solution to this problem, which is developed in [12], is to increase

the given monitor function m — in a procedure that is called “padding” — in such a

way that, when thepaddedmonitor function is equidistributed, theratio of consecutive

grid steps are bounded as required. The padding is chosen so that the equidistributing

grid has adjacent steps with constant ratios equal to the maximum allowed. Such a

function exists and is given by the following formal results [12]:

Let λ be a given number. For any m ∈ C+, we define a padding P(m) of m by

P (m)(z) = maxx∈[xL,xR]

m(x)1 + λ |z − x| m(x)

(2.6)

P(m) has the properties:

1. P(m) is strictly positive on [xL, xR], except in the case m ≡ 0

2. P(m) ≥ m on [xL, xR]3. P(P(m)) = P(m) on [xL, xR]

Let λ > 0, m ∈ C+ and a grid π be equidistributing on [xL, xR] with respect toP(m) and some c > 0. Then

1. the grid π is subequidistributing with respect to m and c

2. for K = eλc we have

1

K≤ xi

xi−1≤ K, i = 2, . . . , N  − 1 .

Based on these results, it is now possible to build a grid which is subequidistributingwith respect to m and c and which is locally bounded with respect to K. In practice,

the algorithm proceeds as follows:

1. pad the monitor function using λ = (log K)/c

2. determine the smallest number of nodes N  such that N c ≥ xR

xLP(m)dx

3. equidistribute P(m) with respect to d  = ( xR

xLP(m)dx)/N 

Clearly, we cannot know the constant d  with respect to which the padded functionP(m) should be equidistributed before actually performing the padding. The proce-

dure could therefore be iterated, padding the monitor function using λ = (log K)/d 

and so on.

As d  ≤ c, the grid is locally bounded with respect to a constant L ≤ K , so that

the number of points in the grid may be greater than required to strictly satisfy the

problem constraints.

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2.2.2 Time-Stepping Procedure and Implementation Details

The adaptive grid refinement is a static procedure and as such, proceeds in four

separate steps:

1. approximation of the spatial derivatives on a fixed nonuniform grid

2. time integration of the resulting semi-discrete ODEs

3. adaptation/refinement of the spatial grid

4. interpolation of the solution to produce new initial conditions

In Step 1, the spatial derivatives are approximated using finite difference approx-

imations up to any level of accuracy on a nonuniform grid as implemented in thestandard Fortran subroutine WEIGHTS by Fornberg [3]. This algorithm is used

for generating “direct” as well as “stagewise” schemes. In the latter case, higher-

order derivatives are obtained by successive numerical differentiations of lower-order

derivatives. An example of the use of these particular schemes will be given in the

section on the Korteweg-de Vries equation.

In Step 2, time integration of the semi-discrete system of stiff ODEs or DAEs is ac-

complished using the variable step, fifth-order, implicit Runge–Kutta solver RADAU5

[7]. Time integration is halted periodically, i.e., every N adapt integration steps, to

adapt/refine the spatial grid (N adapt = 1 corresponds to alternate time integration and

grid adaptation).

In Step 3, the grid is updated using the algorithm described in the previous section.

For this purpose, a new subroutine called AGEREG (from AGE, a routine previously

developedbytheauthors [22], in whichspatial grid regularization isnow incorporated)

has been implemented. The coding of this routine has been inspired largely by the

code NEWMESH that Steinebach developed in his Ph.D. Thesis [24] and by the PDE

software package SPRINT [1]. This algorithm is extended to problems in two space

dimensions in Chapter 6.Implementation issues involve computation of the monitor function (2.3) or (2.4)

using cubic spline differentiators, padding of the monitor function in two sweeps of 

the grid (in the forward and backward direction), and grid equidistribution by inverse

linear interpolation based on a trapezoidal rule.

Finally, in Step 4, the solution is interpolated using cubic splines inorder togenerate

initial conditions on the new grid.

2.3 Application Examples

In the past several years, the solution of partial differential equations governing

nonlinear waves in dispersive media has aroused considerable interest. As a result

numerous research papers dealing with mathematical or numerical aspects of these

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equations as well as several research monographs (see, e.g., [28, 13, 25]) have been

published.

Particularly, the existence of solitary waves or solitons has been a subject of fasci-

nation. Solitons are solutions that possess twopermanence properties: (a) they evolvewithout change of shape over large distances, and (b) they exhibit elastic collisions

with other solitons. Solitons exist in some particular equations such as the nonlinear

Schrödinger (NLS) equation and the Korteweg-de Vries (KdV) equation, which will

be studied numerically in the continuation of this chapter.

The importance of solitons in today’s literature is explained by the fact that very

general nonlinear wave equations have particular regimes in which their long time

behavior is modeled by an equation that has solitons. This modeling procedure,

called the reductive perturbation method, may be compared to a linearization in which

solitons play the role of exponential solutions [13].

2.3.1 The Nonlinear Schrödinger Equation

The nonlinear Schrödinger equation arises in a number of physical situations in

the description of nonlinear waves (see, e.g., [28]) such as the propagation of a laser

beam in a medium whose index of refraction is sensitive to the wave amplitude, the

modulational instability of water waves, the propagation of heat pulses in anharmonic

crystals, and the nonlinear modulation of plasma waves.

It provides a canonical description of the modulation of nearly monochromatic

wavetrains propagating in a weakly nonlinear dispersive medium. When written in a

reference frame moving at the group velocity of the carrying wave, it takes the simple

form

iut  + uxx + qu

u2 = 0, −∞ ≤ x ≤ ∞, t  ≥ 0 (2.7)

u(x, 0) = u0(x) (2.8)

where u=

u(x,t) is a complex-valued function defined over the whole real line and

q is a real constant. In the following, only the focusing case is considered, which

corresponds to q > 0.

In this equation, the cubic term opposes dispersion (i.e., the term in uxx ) and

allows the existence of solutions, such as the solitons, where the competing forces of 

dispersion and nonlinearity balance each other exactly.

Zakharov and Shabat [29] derived analytical solutions to the initial-value problem

(2.7) and (2.8) using an inverse scattering procedure [4]. This IVP possesses an

infinite set of conservation laws, among which the conservation of the energy

E(u) = 

|u|2 dx (2.9)

and the Hamiltonian

H(u) = 

|ux |2 − 1

2q |u|4

dx (2.10)

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plays an important role in the analysis of the NLS equation.

In addition, a number of finite-difference and finite-element schemes have been

suggested for the numerical study of (2.7) and (2.8); see, e.g., [2, 6, 8, 19, 20, 21,

26]. Of particular significance in the development of these schemes is the numericalconservation of the invariants (2.9) and (2.10). Particularly, conservation of energy

implies the L2-boundedness of the solution, thus preventing blow-up of the computed

solution.

For the numerical treatment of the NLS equation, we assume that for the time

interval under consideration, the solution vanishes outside some interval (xL, xR )

and we introduce (artificial) homogeneous Dirichlet boundary conditions

u (xL, t )

=u (xR , t )

=0 (2.11)

thereby converting the pure initial-value problem (2.7) and (2.8) into an initial-

boundary value problem. Note that homogeneous Neumann boundary conditions

ux (xL, t) = ux (xR , t ) = 0 could have been used as well.

Moreover, the complex-valued function u(x,t) is decomposed into its real and

imaginary parts u(x,t) = v(x,t) + iw(x,t) so that (2.7), (2.8), and (2.11) can be

re-written as

vt 

+wxx

+qw v2

+w2 =

0

wt  − vxx − qv

v2 + w2 = 0 (2.12)

v(x, 0) = v0(x)

w(x, 0) = w0(x) (2.13)

v (xL, t ) = v (xR, t ) = 0

w (xL, t ) = w (xR, t ) = 0 . (2.14)

In the following sections, several particular cases, including the propagation of 

a single soliton, the interaction between two solitons, and the bound state of threesolitons, are investigated.

2.3.1.1 Propagation of a Single Soliton

The initial condition is given by

u0(x) = 

2a/q exp [i0.5s (x − x0)] sech√ 

a (x − x0)

(2.15)

and the corresponding soliton solution is

u(x,t) = 

2a/q exp

i0.5s (x − x0) −

0.25s2 − a

t 

sech√ 

a (x − x0) − st 

.

(2.16)

The modulus |u(x,t)| represents a wave initially located in x = x0 traveling with

velocity s in the positive direction of x. The amplitude√ 

2a/q is determined by the

real parameter a.

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As in [20, 21], the problem is solved for q = 1, a = 1, s = 1, and x0 = 0. The time

interval of interest is (0, 30) and, accordingly, the artificial boundaries are located at

xL = −30 and xR = 70. This simple example is used as a first test for our adaptive

mesh refinement algorithm, allowing the effect of the several tuning parameters to behighlighted.

Numerical Results Thesecond-order spatial derivatives in (2.12)are approximated

using a 5-point centered finite-difference scheme. The resulting system of semi-

discrete ODEs is integrated using the implicit RK solver RADAU5, with absolute

and relative error tolerances set to atol = rtol = 10−5.

First, tuning of the adaptive mesh refinement algorithm is accomplished in order to

obtain good numerical accuracy and computational efficiency. A curvature monitor

function (2.4) is used with the tuning parameters α = 10−5, β = ∞ (no limitation

of the second-order derivative), c = 0.1, K = 1.2, and N adapt = 1 (alternate time

integration and grid adaptation). Accuracy is evaluated by computing the L2-norm

of the error in the numerical solution as compared to the exact solution (2.16)

e2 =

1

(xR

−xL)

N −1

i=1

(xi+1 − xi )

2 error (xi )2 + error (xi+1)2

. (2.17)

Figure 2.1 shows the computed solution at time t  = 0, 5, 10, 15, 20, 25, 30 (dots)

and the exact solution (solid line). The location of the adaptivegrid points is displayed

at the bottom of the figure.

FIGURE 2.1

CSE: propagation of a single soliton every 5 units in t  — numerical solution on

an adaptive grid (dots) and exact solution (solid line).

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With a fixed, uniform grid, N  = 651 nodes are required to achieve the same level

of accuracy (but not the same level of graphical resolution since a smaller number of 

nodes are concentrated in the peak, yielding a coarser picture of the soliton). Table 2.1

compares the computational statistics (i.e., the number of function evaluations FNS,the number of Jacobian evaluations JACS, the number of computed steps STEPS,

and the computational costs CPU; for simplicity, the computational costs have been

normalized with respect to the CPU of the adaptive grid solution) when using an

adaptive or a fixed, uniform grid. In the latter case, time integration can be performed

without interruption. However, it is interesting to consider the situation where it is

halted after every time integration step, in order to evaluate the computational costs

associated with the solver restarts (apparently, about 50% more computation time is

required when restarting the solver periodically, resulting however in a slightly better

overall accuracy). Note that the values of e2 reported in Table 2.1 are averagevalues over the time span of interest.

Clearly, the adaptive grid algorithm performs very satisfactorily, both in terms of 

accuracy and computational costs. The number of nodes is almost constant, a logical

observation since the soliton propagates without change of shape.

The effects of the several tuning parameters are now investigated. In this example,

α and β do not play a significant role, i.e., α and β can be set to 0 and

∞, respectively

(a small value of α, as the one selected in our reference run, has a positive effecton the grid regularity). If c is reduced, the number of nodes increases, resulting in

improved accuracy and in an almost proportional increase of the CPU time (which

shows that the grid regularity — which is determined by K — is unaffected). If K is

reduced, grid regularity is enforced, resulting in an increase of the number of nodes

and, in turn, of the CPU time. The influence of c and K on the grid spacing xi is

illustrated in Figures 2.2 and 2.3. The grid adaptation period N adapt can be increased

up to 10, without significant effect on the accuracy, but with a positive effect on the

computation time which is reduced to 0.52. It is possible to further increase N adapt

(up to 40) and obtain a satisfactory numerical solution. However, too infrequent

grid adaptation yields an increase in the number of nodes (to compensate for their

inappropriate placement) and a more difficult time-stepping procedure (characterized

by increasing computation time).

Table 2.1 Propagation of a Single Soliton (CSE): Computational

Statistics and Average Values of the L2-Norm of the ErrorGrid N N adapt FNS JACS STEPS CPU e2

Adaptive 85 − 86 1 2328 420 426 1.0 ≈ 5 × 10−5

Uniform 651 1 1737 312 318 5.0 ≈ 6 × 10−5

Uniform 651 ∞ 1047 144 150 2.7 ≈ 8 × 10−5

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FIGURE 2.2

Influence of c on the grid spacing xi .

FIGURE 2.3

Influence of K on the grid spacing xi .

2.3.1.2 Interaction of Two Solitons

Consider now an initial condition given by

u0(x) = 

2a1/q exp [i0.5s1(x − x01)] sech√ 

a1 (x − x01)

+ 2a2/q exp [i0.5s2 (x

−x02)] sech

√ a2 (x

−x02) (2.18)

which is the superposition of two solitons with amplitudes a1 and a2, respectively,

located in x01 and x02, and traveling with speed s1 and s2.

Specifically, we consider two solitons with different amplitudes and initial loca-

tions, propagating in opposite directions, e.g., a1 = 0.2, a2 = 0.5, x01 = 0, x02 = 25,

s1 = 1.0, s2 = −0.2. The two solitons interact as if they were particle-like entities,

i.e., they exhibit elastic collisions from which they emerge with the same shape. The

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time interval of interest is (0, 45) and, accordingly, the artificial boundaries are located

at xL = −20 and xR = 80.

Numerical Results This example motivates the use of an adaptive grid technique

with a variable number of nodes. Indeed, more nodes are required for reproducing the

two separate solitons traveling in opposite direction than for capturing the interaction

of these two entities. A curvature monitor function (2.4) is used with the tuning

parameters α = 10−5, β = ∞ (no limitation of the second-order derivative), c = 0.1,

K = 1.5, and N adapt = 5. Tolerances atol = rtol = 10−6 are imposed for the time

integrationwith RADAU5. The numberofnodesvariesbetween N  = 89and109over

the time interval (0, 45). Figures 2.4, 2.5, and 2.6 show snapshots at t  = 0, 20, 45,

respectively, and compare the adaptive grid solution (dots) with a reference solution

computed with 2001 fixed nodes (solid line) as well as with a fixed grid solutionwith N  = 140 nodes (dashed line), which requires the same computational expense

as the adaptive grid solution. Clearly, the adaptive grid algorithm performs very

satisfactorily, whereas the fixed grid solution with N  = 140 nodes is unacceptable.

FIGURE 2.4

Two solitons traveling in opposite directions (t  = 0).

Figure 2.7 shows the original curvature-monitor function (2.4) computed on the

initial condition (2.18) and the padded monitor function (2.6).

2.3.1.3 Bound State of Several Solitons

The parameters a and s are independent so that solitons with different amplitudescan move at the same speed, all the time interacting with one another. In the early

1980s, Miles [15] showed that for q = 2N 2 (where N  is a positive integer) and an

initial condition given by

u0(x) = sech(x) (2.19)

u(x,t) is a bound state of N  solitons.

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FIGURE 2.5

Interaction of two solitons (t  = 20).

FIGURE 2.6

Two solitons after an elastic collision (t  = 45).

The bound state of several solitons results in very steep gradients in space and

time and provides a more severe test of our numerical scheme than the two previous

situations. As in [8, 20, 21], we consider the bound state of three solitons, i.e., q = 18,

and use artificial boundaries located at xL = −20 and xR = 20. In this problem, thesolution is periodic in time, a period being approximately T  = 0.8. In the following

numerical investigations, we are particularly interested in the integration of (2.7) and

(2.19) over large time intervals. Indeed, results reported in previous studies (see,

e.g., [8, 22]) show that accuracy deteriorates as time evolves [phase errors, non-

conservation of the invariants (2.9) and (2.10), spurious oscillations, non-symmetric

profiles, and eventually blow-up of the numerical solution].

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FIGURE 2.7

Monitor function m (solid line) and padded monitor function P(m) (dotted line)

(t  = 0).

Numerical Results A curvature monitor function (2.4) is used with the tuning

parameters α = 2.5 × 10−5, β = ∞ (no limitation of the second-order derivative),

c=

0.07, K=

1.5, and N adapt

=5. Tolerances atol

=rtol

=10−5 are imposed for

the time integration with RADAU5.

Over a period (0, 0.8), the number of nodes varies between N  = 66 and 157,

depending on the profile complexity and sharpness. To see what happens in longer

time integrations, the problem is solved over the interval (0, 4), i.e., over 5 periods.

The quality of the numerical solution can be checked by graphical inspection (see

Figure 2.8 where the numerical results are graphed every 0.2 units of time) and

by monitoring the conservation of the invariants (2.9) and (2.10), which should be

close to their exact values E = 2 and H  = 2/3(1 − q) = −34/3 = −11.333.

FIGURE 2.8

Bound state of three solitons from t  = 0 to t  = 4 at time intervals of 0.2.

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Indeed, Figure 2.9 shows that the approximations to these quantities (computed by

numerical quadrature) are very well conserved (the average values are_

E = 2.003

and_

H 

= −11.403).

FIGURE 2.9

Evolution of the conserved quantities E (top) and H  (bottom).

From Figure 2.8, the periodic behavior is observed throughout the time interval

(0, 4), but the numerical solution suffers from phase errors. For instance, the solution

profile computed at t  = 0.6 should be reproduced at t  = 1.4, 2.2, and 3.8. However,

Figure 2.10, which compares the profile at t  = 0.6 (solid line) with the profiles

computed at t  = 3.73, 3.74, 3.75, 3.76, respectively (dotted lines), shows that this

profile is reproduced at t  = 3.75 (i.e., with a phase error of 0.05 unit of time or

22.5◦). It is important to note that these phase errors are twice as big (e.g., 45◦)

when a solution is computed on a fixed, uniform grid with N  = 3001 nodes, which

demonstrates the superiority of the adaptive grid refinement method.

2.3.2 The Derivative Nonlinear Schrödinger Equation

The derivative nonlinear Schrödinger equation

iut  + uxx + i

u

u2

x= 0, −∞ ≤ x ≤ ∞, t  ≥ 0 (2.20)

u(x, 0) = u0(x) (2.21)

was originally derived by Mjolhus [16] to describe the long wavelength propagationof circular polarized waves parallel to the magnetic field in a cold plasma. In (2.20),

u(x,t) = v(x,t) + iw(x,t) represents the transverse components of the magnetic

field to lowest order. The time and space coordinates t , x are in a reference frame

traveling with the Alvén speed. Using the inverse scattering technique, Kaup and

Newell [11] obtained the one-soliton solution and demonstrated the existence of an

infinity of conservation laws.

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FIGURE 2.10Phase error evaluation: comparison between solution profile at t  = 0.6 and

computed profiles at t  = 3.73, 3.74, 3.75, and 3.76.

Here, we investigate numerically a simple example corresponding to an initial

condition in the form

u0(x)

=sech(x) . (2.22)

2.3.2.1 Numerical Results

As the sech-initial conditions (2.22) disperse away, the adaptive grid algorithm

has to gradually add nodes in spatial regions further away from the initial location.

The time span of interest is (0, 50) and, accordingly, artificial homogeneous Dirichlet

boundary conditions are imposed at xL = −300 and xR = 500.

Figures 2.11 and 2.12 show the transverse component profiles v(x,t) and w(x,t) at

t  = 0, 5, 10, 15 computed on an adaptive grid based on a curvature monitor function

with the tuning parameters α = 10−4

, β = ∞, c = 0.05, K = 1.1, and N adapt = 5.Tolerances atol = rtol = 10−5 are imposed for the time integration with RADAU5.

The evolution of the solution modulus |u(x,t)| is graphed every 5 units in t , along

with the location of the adaptive grid points, in Figure 2.13. On the time interval

(0, 50), the number of nodes gradually increases from N  = 276 to 2831. In this case,

the main advantage of a refinement procedure over a fixed uniform grid solution is to

allow, at any time, a fine description of the solution details [for instance, the initial

condition is a very narrow peak, which would require a very large number of nodes

on the complete space interval (

−300, 500) to be represented accurately].

2.3.3 The Korteweg-de Vries Equation

This equation was originally introduced by Korteweg and de Vries in 1895 [14] to

describe thebehaviorofsmall amplitudeshallow-water waves inonespace dimension.

Over the years, the KdV equation has found application in several areas, including

plasma physics, liquid-gas bubble mixtures, and anharmonic crystals.

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FIGURE 2.11DNLS equation: graph of the transverse component v(x,t) at t  = 0, 5, 10, 15.

FIGURE 2.12

DNLS equation: graph of the transverse component w(x,t) at t  = 0, 5, 10, 15.

In this section, attention is focused on the classical KdV equation given by

ut  + 6uux + uxx x = 0 − ∞ ≤ x ≤ ∞, t  ≥ 0 (2.23)

u(x, 0) = u0(x) (2.24)

which combines the effect of nonlinearity and dispersion. The spectral approach (or

inverse scattering transform [4]) has had a major impact on the analysis of the KdV

equation. This approach can be used to produce analytical solutions as well as to

develop numerical algorithms; see for instance the surveys of Taha and Ablowitz [27]and Nouri and Sloan [17].

The IVP (2.23) and (2.24) possesses an infinity of invariants, e.g., the conservation

of mass

I 1(u) = 

u dx (2.25)

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FIGURE 2.13

DNLS equation: evolution of the modulus of the solution every 5 units in t .

the conservation of energy

I 2(u)

=  u2 dx (2.26)

and a conservation law proposed by Whitham [28]

I 3(u) = 

2u3 − u3x

dx . (2.27)

In the following, the propagation of a single soliton

u(x,t) = 0.5 s sech2

0.5√ 

s(x − st ) (2.28)

is investigated numerically. Particularly, the importance of appropriate finite differ-

ence approximations for the dispersive term uxx x in (2.23) is highlighted.

2.3.3.1 Numerical Results

The KdV equation is solved for an initial condition given by (2.28) with s = 0.5,

i.e.,

u0(x) = 0.25 sech20.53/2x

For the time span under consideration, e.g., 0 ≤ t  ≤ 70, it is assumed that the solution

vanishes outside a finite interval [−30, 70]. At the endpoints xL = −30 and xR = 70,

homogeneous Dirichlet boundary conditions are imposed, so that the pure IVP (2.23)

and (2.24) is converted into an IBVP.

One of the main difficulties encountered in the MOL solution of the KdV equa-

tion is the approximation of the dispersive term uxxx , which appears as a primary

determinant of the solution accuracy. In a previous work [23], the authors observed

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that higher-order finite difference schemes for the third-order spatial derivative (e.g.,

5-, 7-, 9-, or 11-point centered schemes), which perform satisfactorily on a uniform

spatial grid, give poor results on a nonuniform, adaptive grid. Instead, lower-order

stagewise difference schemes, e.g., successive numerical computation of a first-orderderivative uxx x = ((ux )x )x , produce very satisfactory solutions. Further numerical

investigations confirm these observations and show that, for long integration times

[the results presented in [23] were restricted to integrating over a relatively short

time interval (0, 30)], simulation runs using higher-order finite difference schemes

eventually fail, even on a fixed uniform grid. Hence, stagewise differentiation is used

throughout the present study (the reader interested in the stability and convergence

analysis of specific schemes for third-order derivative terms is referred to [5]). The

resulting system of semi-discrete ODEs is integrated using the implicit RK solver

RADAU5, with absolute and relative error tolerances set to atol = rtol = 10−5.

Very satisfactory numerical results can be obtained using either an arc-length mon-

itor function (2.3) with α = 0, β = ∞, c = 0.005, K = 1.1, and N adapt = 1, or

curvature monitor function (2.4) with α = 10−5, β = ∞, c = 0.02, K = 1.1, and

N adapt = 1. In both cases, grid regularity is enforced using a relatively small value for

K = 1.1 (remember that K = 1 corresponds to a uniform grid). This constraint on

the grid regularity is related to the delicate approximation of the third-order derivative

term. In contrast with the discussion on the effect of K when studying the CSE equa-

tion, numerical experiments show that if K is increased, the computation time doesnot decrease with the number of nodes, indicating that grid distortion is detrimental

to the time-stepping procedure.

Figures 2.14 and 2.15 show how differently the nodes are distributed according

to these two monitor functions. To achieve similar accuracy, a uniform grid with

N  = 701 nodes is required. Tables 2.2 and 2.3 give the computational statistics

and several indicators of the solution quality, i.e., the L2-norm of the error (2.17)

(average values over the time span of interest) and the values of the invariants at the

final time t 

=70 (for the example under consideration, the exact [up to 5 decimal

places] values of the invariants are I 1 = 1.41421, I 2 = 0.11785, I 3 = 0.7071). In

this application example, the adaptive grid solution does not provide much benefit in

terms of computational expense, but allows a finer graphical resolution of the solitary

wave.

Table 2.2 Propagation of a Single Soliton (KdVE):

Computational Statistics

Grid N N adapt FNS JACS STEPS CPU

Adaptive (2.3) 153 − 154 1 965 187 197 1.0Adaptive (2.4) 145 − 147 1 1051 249 259 1.1

Uniform 701 1 720 128 135 2.5Uniform 701 ∞ 452 59 65 1.5

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FIGURE 2.14

KdV equation: propagation of a single soliton every 10 units in t : numerical

solution on an adaptive grid based on an arc-length monitor (dots) and exact

solution (solid line).

FIGURE 2.15

KdV equation: propagation of a single soliton every 10 units in t  — numerical

solution on an adaptive grid based on a curvature monitor (dots) and exact

solution (solid line).

2.3.4 The Korteweg-de Vries-Burgers EquationJohnson [10] derived the Korteweg-de Vries-Burgers (KdVB) equation in the study

of the weak effects of dispersion, dissipation, and nonlinearity in waves propagating

in a liquid-filled elastic tube. The KdVB equation for u(x,t) is given by

ut  + 2auux + 5buxx + cuxx x = 0 − ∞ ≤ x ≤ ∞, t  ≥ 0 (2.29)

u(x, 0) = u0(x) . (2.30)

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Table 2.3 Propagation of a Single Soliton (KdVE): Conserved Quantities

and Average Values of L2-Norm of the Error

Grid N I 1

I 2

I 3

e2

Adaptive (2.3) 153 − 154 1.41217 0.11746 0.07033 ≈ 1.2 × 10−4

Adaptive (2.4) 145 − 147 1.41563 0.11808 0.07099 ≈ 1.4 × 10−4

Uniform 701 1.41292 0.11784 0.07076 ≈ 1.2 × 10−5

In the limits b → 0 or c → 0, i.e., when the effects of dissipation or dispersion are

negligible, the KdVB equation reduces either to the KdV equation

ut  + 2auux + cuxx x = 0 (2.31)

(which has been investigated numerically for a = 3 and c = 1 in the previous section)

or the well-known Burgers equation

ut  + 2auux + 5buxx = 0 (2.32)

In [9], Jeffrey and Xu introduced a transformation that reduces the KdVB equation

toa quadratic formwhich can besolved in terms ofa seriesofexponentials. Incontrast

with the KdV equation, which possesses an infinite sequence of exact solutions (the n-soliton solutions), the quadratic form of the KdVB equation yields only two traveling

waves given by

u1(x,t) = 3b2

2ac

sech2

ϑ

2

+ 2 tanh

ϑ

2

+ 2

(2.33)

with ϑ = bc

x −

6b3

c2

t  + β and

u2(x,t) = 3b2

2ac

sech2

ϑ

2

− 2 tanh

ϑ

2

− 2

(2.34)

with ϑ = − bc

x −

6b3

c2

t  + β.

These solutions cannot be reduced to the sech2 solution to the KdV equation in the

limit b → 0 or to the tanh solution to Burgers equation in the limit c → 0.

2.3.4.1 Numerical Results

The problem is solved for a = 1, b = −1, c = 3, and β = 0 and an initial

condition corresponding to (2.33). The time interval of interest is (0, 15) and, ac-

cordingly, the artificial boundaries are located in xL = −15 and xR = 100. At the

endpoints xL = −15 and xR = 100, homogeneous Dirichlet boundary conditions

are imposed, so that the pure IVP (2.29) and (2.30) is converted into an IBVP. As

for the classical KdV equation, stagewise differentiation for the approximation of the

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third-order derivative term yields better performance than direct differentiation using,

e.g., a 7-point centered finite difference scheme. Tolerances atol = rtol = 10−4 are

imposed for the time integration with RADAU5.

The best results, in terms of accuracy and computational expense, are obtainedwith a curvature monitor function and the following parameters: α = 0, β = ∞,

c = 0.03, K = 1.1, and N adapt = 1. The number of nodes is almost constant, i.e.,

N  ≈ 141. The solution is graphed at time t  = 0, 3, 6, 9, 12, 15 in Figure 2.16.

FIGURE 2.16

KdVB equation: numerical solution on an adaptive grid based on a curvature

monitor (dots) and exact solution (solid line) graphed every 3 units in t .

2.3.5 KdV-Like Equations: The Compactons

Seeking to understand the role of nonlinear dispersion in the formation of patterns

in liquid drops, Rosenau and Hyman [18] introduced a family of fully nonlinear

KdV-like equations in the form

ut  +

um

xx+

un

xxx= 0, m > 0, 1 < n ≤ 3 . (2.35)

These equations, which are denoted K(m, n), have the property that for certain m

and n, their solitary wave solutions have compact support. This remarkable propertyhas suggested the name “compacton” to these authors.

In fact, nonlinear dispersion in (2.35) (which is accounted for by the term (un)xx x)

is weaker for small u than linear dispersion in the classical KdV equation, which

allows the formation of a compact-support solution. On the other hand, dispersion is

much more important at high amplitudes and counterbalances the steepening effect

of nonlinear convection.

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Particularly, the K(2, 2) equation possesses a solitary wave solution with a compact

support given by

uc(x,t) =4s

3 cos2(x

−st 

4 ), |x − st | ≤ 2π

= 0, otherwise . (2.36)

Although the second derivative of the compacton is discontinuous at its edges,

(2.36) is a strong solution of Equation (2.35) since the third derivative is applied to

u2, which has three smooth derivatives everywhere including the edge.

The compacton’s amplitude depends on its speed but, unlike the KdV-soliton which

narrows as the speed increases, its width is independent of the speed.

Similarly to the soliton interactions associated with the cubic Schrödinger equationor the classical Korteweg-de Vries equation, compactons exhibit elastic collisions, in

which, after colliding with other compactons, they emerge with the same coherent

shape. However, the point where two compactons interact is marked by the birth of 

a low amplitude compacton-anticompacton pair.

2.3.5.1 Numerical Results

First, attention is focused on the propagation of a single compacton (2.36) with

speed s

=0.5 over the time interval (0, 80). Accordingly, homogeneous Dirichlet

boundary conditions are imposed at xL = −30 and xR = 70.

As stressed in [18], there are several numerical difficulties in solving the K(2, 2)

equation, which are due to nonlinear dispersion and the lack of smoothness at the

edge of the compacton, possibly leading to instability.

Using the numerical methods described in the previous sections, it was not possible

to solve satisfactorily the K(2, 2) equation on a fixed uniform grid, with the exception

of the particular setting:

• stagewise differentiation of the nonlinear dispersive term

• N  = 501 spatial nodes

• time integration with atol = rtol = 10−5

Even in this fortuitous situation, the graph of the solution (chat 2.17) displays

unacceptable downstream oscillations. However, any attempt to improve on this

situation by increasing the number of nodes or reducing the error tolerances leads to

failure of the simulation run.

When using the adaptive grid procedure, only the arc-length monitor functionallows the problem to be solved (i.e., every attempt to solve this problem with a

curvature monitor function failed). The tuning parameters take the following values:

α = 10−6, β = ∞, c = 0.01, K = 1.1, and N adapt = 1. The corresponding solution,

which is now very satisfactory, is graphed every 10 units in t  in Figure 2.18. In fact,

with thesame tuningparameters, it appears even possible to approximate thenonlinear

dispersive term with a classical 7-point centered finite-difference scheme (instead of 

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using stagewise differentiation, which seems to be a “universal” solution for all the

KdV-like problems considered thus far). In this latter case, accuracy improves at

the price of larger computational costs. The computational statistics as well as the

average value of the L2-norm of the error are summarized in Table 2.4.

FIGURE 2.17

Propagation of a single compacton every 10 units in t : numerical solution on a

fixed uniform grid with N  = 501 nodes (dots) and exact solution (solid line).

FIGURE 2.18

Propagation of a single compacton every 10 units in t : numerical solution on

an adaptive grid based on an arc-length monitor (dots) and exact solution (solid

line).

Second, the interaction of two compactons initially centered in x01 = 0 and x02 =15, respectively, and traveling in the same direction but with different speeds s1 = 0.5

and s2 = 0.25, is considered. The time span of interest is now (0, 120). As time

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Table 2.4 Propagation of a Single Compacton: Computational Statistics andAverage Values of the L2-Norm of the Error

Grid N  Approx. N adapt FNS JACS STEPS CPU

e

2

Adaptive 203 − 204 ((ux )x )x 1 1813 228 448 1.0 ≈ 1.5 × 10−3

Adaptive 203 − 204 uxx x 1 3258 513 798 2.3 ≈ 3.5 × 10−4

Uniform 501 ((ux )x )x 1 1514 367 374 2.5 ≈ 1.2 × 10−3

evolves, the faster compacton catches the slower one and passes through it. The

point where the two compactons collide is marked by the birth of a low amplitude

compacton-anticompacton pair.

All our efforts to solve this challenging problem on a fixed uniform grid wereunsuccessful, and only an adaptive grid solution based on an arc-length monitor

function could be obtained, after quite a lot of tuning, for the parameter setting:

• stagewise differentiation of the nonlinear dispersive term

• α = 10−3, β = ∞, c = 0.015, K = 1.5, and N adapt = 1

• time integration with atol = rtol = 10−4

In contrast with our previous experiments with the KdV equation, a large value of 

α is used to force a relatively high and almost uniform density of nodes outside the

compactons, whereas a large value of K allows grid deformations and higher node

concentrations in the compactons. The solution is graphed every 10 units in t  in

Figure 2.19.

FIGURE 2.19

Interaction between two compactons every 10 units in t .

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2.4 Conclusions

In this chapter, a simple static grid refinementalgorithm is implemented and applied

to a set of nonlinear dispersive wave problems. The number and placement of the

nodes is determined by equidistributing a monitor function related to the arc-length

or the curvature of the computed solution. As grid distortion is detrimental to spatial

accuracy and stiffness of the semi-discrete system of ordinary differential equations,

spatial grid regularization is accomplished by padding the monitor function. This

procedure originally devised by Kautsky and Nichols [12] enforces that the ratio

between adjacent grid steps lies between two bounds specified by the user.

First, some solutions of the nonlinear Schrödinger equation, including the propa-gation of a single soliton, the interaction between two solitons traveling in opposite

direction, and the bound state of three solitons, are studied. In all these test-examples,

very satisfactory numerical solutions, both in terms of accuracy and computational

demand, can be obtained. Particularly, long time integration of the bound state of 

three solitons shows excellent conservation of invariants (energy and Hamiltonian)

as well as relatively small phase errors. In addition, some numerical results for the

derivative nonlinear Schrödinger equation are presented.

Second, several KdV-like equations, including the classical Korteweg-de Vriesequation, the Korteweg-de Vries-Burgers equation, and a fully nonlinear KdV equa-

tion giving rise tocompactons, areconsidered. There areseveralnumerical difficulties

in solving these equations, which are related to the approximation of the third-order

spatial derivative term (dispersive term). Somewhat surprisingly, high-order “direct”

finite difference schemes provide relatively poor results, whereas low-order “stage-

wise” schemes (which proceed by successive numerical differentiation of lower order

derivatives) appear as simple and efficient approximations in most cases.

The propagation and interaction of compactons are very challenging problems, for

which numerical solutions could only be obtained at the price of a careful selectionof the algorithm parameters. At this stage, further investigations are required.

References

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and 89).

[2] M. Delfour, M. Fortin, and G. Payne, Finite difference solution of a nonlinear

Schrödinger equation, J. Comp. Phys., 44, (1981), 277–288.

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[3] B.Fornberg, Generation offinite difference formulasonarbitrarily spacedgrids,

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[20] J.M. Sanz-Serna and J.G. Verwer, Conservative and nonconservative schemes

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13–25.[24] G. Steinebach, Die Linienmethode und ROW-Verfahren zur Abfluss- und 

Prozess-simulation in Fliessgewässern am Beispielvon Rhein und Mosel, Ph.D.

Thesis, Darmstadt Technical University, Germany, 1995.

[25] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation — Self-

Focusing and Wave Collapse, Springer-Verlag, Berlin, 1999.

[26] T.R. Taha and M.J. Ablowitz, Analytical and numerical aspects of certain non-

linear evolution equation. II. Numerical, nonlinear Schrödinger equation, J.Comp. Phys., 55, (1984), 203–230.

[27] T.R. Taha and M.J. Ablowitz, Analytical and numerical aspects of certain non-

linear evolution equation. III. Numerical, nonlinear Korteweg-de Vries equa-

tion, J. Comp. Phys., 55, (1984), 231–253.

[28] G.B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York,

1974.

[29] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing

and one-dimensional self-modulation of waves in nonlinear media, Soviet Phys.

 JEPT, 34, (1972), 62–69.

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Chapter 3

 Numerical Solutions of the Equal WidthWave Equation Using an Adaptive Method of 

 Lines

S. Hamdi, J.J. Gottlieb, and J.S. Hansen

Abstract The equal-width wave (EW) equation is a model partial differential equa-

tion for the simulation of one-dimensional wave propagation in media with nonlinear

wave steepening and dispersion processes. The background of the EW equation is

reviewed and this equation is solved by using an advanced numerical method of lineswith an adaptive grid whose node movement is based on an equidistribution princi-

ple. The solution procedure is described and the performance of the solution method

is assessed by means of computed solutions and error measures. Many numerical

solutions are presented to illustrate important features of the propagation of a solitary

wave, the inelastic interaction between two solitary waves, the breakup of a Gaussian

pulse into solitary waves, and the development of an undular bore.

3.1 Introduction

Wave propagation has intrigued scientists for many centuries owing to their fasci-

nating nonlinear behavior, originating with mankind’s observation of the spectacular

breaking of water waves. Some waves can propagate with constant shape and speed,

others when perturbed can undergo decay and shed a trailing disturbance, some can

partially disintegrate and shed a train of weak trailing waves, whereas others can

accelerate as they become spatially narrower and blow-up in amplitude. Such wave

behavior is illustrated in Figure 3.1, where different waves are combined in one time-

distance diagram.

Nonlinear wave phenomena has been studied extensively in recent years by many

researchers, and some of them have directed their efforts at formulating mathematical

models for thedescription of wave propagation in media with nonlinear wave steepen-

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FIGURE 3.1

Illustrations of the behavior of different solitary waves.

ing and dispersion effects. The well-known Korteweg and de Vries (KdV) equation,

ut  + uux + uxx x = 0, is the first classical nonlinear partial differential equation

(PDE) that has been very successful in this regard. This model equation, formulated

by Korteweg and de Vries [19] in the year 1895, simulates the time-dependent motion

of shallow water waves in one space dimension. The pioneering study by Kortewegand de Vries showed that when nonlinear wave steepening, from the nonlinear term

uux , is balanced by wave dispersion, owing to the term linear uxx x , their equation

predicts a unidirectional solitary wave, that is a pulse which moves in one direction

with a permanent shape and constant speed. For example, see the waves labeled (a)

and (b) in Figure 3.1. A remarkable property of these solitary waves is that they can

be exceptionally stable while traveling relatively long distances without undergoing

any noticeable alterations in shape, amplitude, and speed.

Nonlinear wave steepening and dispersion processes are important not only in

hydrodynamics but also in many other disciplines of engineering and science, inwhich the KdV equation has also become a powerful tool for the modeling of wave

phenomena. The study of Berezin and Karpman [4] contains several examples of 

the propagation of nonlinear waves with moderately large wavelengths and small but

finite amplitudes in liquids, compressible gases, cold plasmas, and other media with

dispersive effects.

Benjamin et al. [3] advocated that the PDE ut  + uux + ux − µuxx t  = 0 modeled

the same physical phenomena equally well as the KdV equation, given the same as-

sumptions and approximations that originally led Korteweg and de Vries [19] to their

equation. This PDE of Benjamin et al. [3] is now often called the regularized long

wave (RLW) equation, although it is also known as the BBM equation. The word

regularized in RLW stems from past developments of more expedient mathematical

tools and properties for the RLW equation, as compared to the KdV equation, which

have facilitated rigorous proofs of the existence and uniqueness of periodic and non-

periodic solutions on an unbounded domain, and which have helped prove that these

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solutions are stable and continuous with regard to various initial conditions, including

the perturbation of an initially specified solitary wave.

Peregrine [23] was the first to solve the RLW equation to describe the development

of undular bores, which are smooth solitary waves that were observed propagatingin shallow water channels. The RLW equation was solved successfully in additional

applications involving the time-dependent motion of one-dimensional drift waves in

plasmas, and Morrison et al. [21] mention that the RLW equation can also be used to

simulate Rossby waves for geophysical applications.

Morrison et al. [21] proposed the one-dimensional PDE, ut  + uux − µuxx t  = 0, as

an equally valid and accurate model for the same wave phenomena simulated by the

KdV and RLW equations. This PDE is now called the equal-width (EW) equation

because the solutions for solitary waves with a permanent form and speed, for a given

value of the parameter µ, are waves with an equal width or wavelength for all wave

amplitudes. The EW equation is a simpler and lesser known alternative to the RLW

equation, and the solitary wave solutions are less general because of the equal-width

constraint.

The properties of solutions from the KdV, RLW, and EW equations can differ re-

markably, even though they are model equations for similar types of wave motion.

The solution of the KdV equation for a solitary wave that is initially perturbed illus-

trates that this wave can propagate without significant change in shape and speed and

remain stable, as shown by the waves labeled (a) and (b) in Figure 3.1, whereas the so-lutions of the RLW and EW equations show that a perturbed solitary wave can evolve

instead into a decaying wave with an oscillating tail or evolve into a contracting wave

with amplitude blow-up, as depicted by the waves labeled (c) and (d) in Figure 3.1.

The solution of the KdV equation for the overtaking of one solitary wave by another

features two transmitted waves that retain their original shapes and speeds, but they

are displaced from their original straight trajectories, as depicted by the wave system

labeled (a) in Figure 3.2. Such interactions are called elastic or clean interactions. In

contrast, the solutions of the RLW and EW equations for solitary wave interactions

exhibit transmitted waves that can shed trailing disturbances, can split into a set of weak waves, or can increase unboundedly in speed and amplitude, as shown by the

wave system labeled (b) in Figure 3.2. These latter interactions are labeled inelastic,

anelastic, or unclean interactions.

The KdV equation can be solved by analytical means for some specific problems

and in general by the inverse scattering transform (IST) technique and spectral meth-

ods (SMs). The RLW and EW equations cannot be solved by the IST, but these

equations, as well as the KdV equation, can be solved by using various numerical

techniques (e.g., the method of lines). The differences in the solution procedures

(nonexistence of an IST solution) and related numerical difficulties in solving the

RLW and EW equations in contrast to the KdV equation, all stem directly from the

dispersive term uxx t  in the RLW and EW equations. Mathematically, the KdV equa-

tion is said to be integrable or a completely integrable Hamiltonian system (with

reversible energy exchange between the degrees of freedom), whereas the RLW and

EW equations are nonintegrable, which corresponds directly to their inelastic or un-

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FIGURE 3.2

Illustrations of elastic and inelastic interactions of solitary waves.

clean behavior involving wave interactions. An integrable equation admits an infinite

number of conservation laws and invariants of motion, whereas a nonintegrable equa-

tion has only a limited number of invariants of motion. An integrable equation can

be rewritten as a compatibility condition in terms of two linear equations called theLax pair; see the book by Whitham [32] for details. This last property is the essence

of the IST method.

Distinguishing between solitary waves and solitons is sometimes important. Soli-

tons are very special types of solitary waves that have the property of elastic wave

interactions. Solitary waves associated with integrable equations, such as the KdV

equation, are solitons, whereas those associated with nonintegrable equations, such

as the RLW and EW equations, are not. This restricted definition is adopted herein.

Note that no distinction is made between solitons and solitary waves in plasma physics

and quantum mechanics.From an historical perspective, theRLW andEWequations were originally believed

to be integrable and yield elastic wave interactions, as mentioned by Santarelli [26]

and Abdulloev et al. [1]. This belief stemmed from computations using inaccurate

or low-resolution numerical methods, from which the small effects of inelastic wave

interactions were undetected in the numerical solutions. For example, the slow decay

of a solitary wave and its shedding of weak trailing waves were simply unresolved, or

they may have been misinterpreted or confused as numerically generated oscillations.

This numerical difficulty was first identified and overcome by Santarelli [26] in his

studies of the one-dimensional collision of two solitary waves that produced large and

readily observable inelastic effects in the form of a train of solitary waves between

the transmitted waves. Santarelli’s discoveries were confirmed later by Lewis and

Tjon [20] for similar types of problems.

Accurate numerical techniques for solving the KdV, RLW, and EW equations are

required to realistically capture important large and small qualitative and quantitative

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features of the solution, especially when overtaking and colliding solitary waves

exhibit very rapid solution variations (e.g., at shocks). Several numerical studies

yielding more accurate solutions of the RLW and EW equations, and some other

closely related PDEs, have been reported by Jain et al. [18] and Bona et al. [5].The methods they reviewed are based on classical finite-difference and finite-element

techniques with an accurate space discretization that is normally coupled to a low

second-order time integration scheme; see Gardner et al. [10, 11] for some examples.

The primary weakness of these earlier numerical methods is the use of a low-order

time integration scheme for the solution of long-duration evolutionary waves. A

further shortcoming stems from the use of a simple uniform grid that limits the spatial

resolution of rapidly varying solutions in space.

More accurate numerical techniques have been developed recently for solving

the KdV equation, and these have been implemented in the numerical method of lines (MOL) by, for example, Schiesser [30]. High-order spatial discretizations with

finite-difference schemes are readily usable today with either uniform or nonuni-

form grids by incorporating, for example, the algorithm called WEIGHTS from Forn-

berg [8]. High-order time integration techniques for non-stiff differential equations

are also readily available; for example, the time integrator or solver called RKF45

uses an explicit variable fourth- and fifth-order Runge-Kutta method, as described by

Forsythe et al. [9]. For numerically stiff and explicit systems of ordinary differential

equations, several efficient time integrators are also available; for example, Hairer

and Wanner [14] developed a variable time-step, fifth-order, implicit Runge-Kuttasolver called RADAU5. For numerically stiff and implicit systems of differential-

algebraic equations, Petzold [24] and Hindmarch [17] developed the time integrators

called DASSL and LSODI, respectively, which are both based on variable time-step,

variable-order, backward-differentiation formulae. Adaptive grid techniques, with

node movement based on an equidistribution principle, have become instrumental in

helping to accurately resolve very steep solution gradients and curvatures in space,

and various techniques and programs are now available from a number of authors

such as White [31], Sanz-Serna and Christie [27], Revilla [25] and, more recently,

Saucez et al. [28, 29], who use the MOL approach.This study focuses primarily on solving the EW equation, ut  + uux − µuxx t  = 0,

by using an advanced MOL with adaptive gridding. The main numerical difficulty in

solving this equation stems from thedispersive term uxx t  which results in the coupling

of the space and time derivatives. The spatial discretization produces a fully implicit

set of differential-algebraic equations, and these are integrated numerically in time by

usinga recently developed robust integration solver such asDASSL from Petzold [24].

Our adaptive MOL approach stems originally from the studies of Schiesser [30] and

more recently from our related studies with the KdV and EW equations as reported

by Hamdi et al. [15, 16]. In this study, our numerical techniques are described, the

performance of our methods are assessed by means of numerical results and error

measures, and many interesting problems involving the EW equation are solved and

their solutions presented to illustrate salient features of the propagation of a solitary

wave, the inelastic interaction of two solitary waves, the breakup of a Gaussian pulse

into solitary waves, and the development of an undular bore.

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3.2 Equal-Width Equation

Basic information andanalytical tools for the equal-width (EW) andrelated regular-

ized long wave (RLW) equations are provided in this section. These are prerequisites

for describing the solution procedure, assessing the solution method, and understand-

ing important features of the numerical results.

The EW equation is a partial differential equation (PDE) given by

ut 

+uux

−µuxx t 

=0 , (3.1)

in which u = u(x,t) is a function of the two independent variables x and t  that

normally denote space and time, respectively. As subscripts on u, x and t  denote

partial derivatives of the dependent variable u. The parameter µ is a positive real

constant. In most fluids related problems, u(x,t) represents the wave amplitude

or some similar physical quantity, whereas in plasma applications it is the negative

electrostatic potential. In most applications the terms uux and uxx t  produce nonlinear

wave steepening and dispersion, respectively.

The EWequationisa simpler formof the RLW equation, ut +uux+ux−µuxx t  = 0,

as mentioned in the introduction. The RLW equation is a PDE that has been used to

simulate wave motion in media with nonlinear wave steepening and dispersion, such

as shallow water waves and ion acoustic plasma waves. However, the simpler EW

equation is an equally valid and accurate model for the same wave phenomena; see

the study of Morrison et al. [21] for more details. Although the EW equation can be

transformed into the RLW equation by means of uEW  → uRLW  +1, a solution of the

EW equation cannot provide a solution to the RLW equation because the boundary

conditions are incompatible.The EW equation requires boundary conditions for solitary and other wave motion

of the form u(x,t) → uL and uU  as x → −∞ and +∞, at which the constants uL and

uU  are normallyzero. The boundary conditions for our solutions are approximated on

the finite computational domain xL ≤ x ≤ xU  by u(xL, t) = uL and u(xU , t) = uU ,

which have been used in previous studies. These are good approximations because

our numerical solutions are computed when all of the initial conditions and wave

motion are well within the interior of the domain, such that the amplitudes of the data

and waves die out asymptotically to a constant or zero at each domain boundary.

Some important analytical solutions for the motion of solitary waves for both the

EW and RLW equations are available in the paper of Morrison et al. [21]. These are

given by

u(x,t) = 3 c sech2[k (x − x0 − νt )] , (3.2)

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for which the wave number k and wave speed ν are defined by

k =  1

4µ

for EW, 1

4µ

c

c + 1for RLW,

ν = c for EW,c + 1 for RLW,

(3.3)

for the two equations. These solutions correspond to solitary waves moving in the

positive or negative x directions, depending on the sign of ν. These waves have a

positive or negative constant peak amplitude 3c, unchanging wave shape or profile,

and steady wave speed ν. The wave is centered initially at the location x0. The motion

of such a solitary wave is depicted by the wave labeled (a) in Figure 3.1. Note that

Equation (3.2) can be derived by assuming that a solitary wave of constant shape and

speed exists, having the generic form u(x,t) = f (x − x0 − νt). This form is then

substituted into the RLW and EW equations (PDEs), and the solutions of the resulting

ODEs yield Equation (3.2) for the solitary wave.

Solitary wave solutions of the EW equation exist for all wave speeds −∞ < ν =c < ∞. This is unlike the case of the RLW equation for which solitary wave speeds

exist only when ν = c + 1 < 0 or ν = c + 1 > 1, conditions that make k a real

nonzero number. Alternately, the forbidden wave speeds are 0 ≤ ν ≤ 1 for the RLW

equation.

The solitary wave solution given by Equation (3.2) features a symmetric wave

profile about the path x = x0 + νt , along which the wave has the peak amplitude

upeak  = 3c. To obtain the width λ of this wave at a given fraction of the peak ampli-

tude, consider another parallel path x = x0 + νt  + λ/2, along which the amplitude

is also constant and given by uλ = 3c sech2[kλ/2]. A normalized amplitude can be

defined by u = uλ/upeak  = sech2[kλ/2], where u is a specified constant (e.g., 1/2).

The solution for the previously defined width is then

λ = 2

ksech−1

√ u

= 4√ µ sech−

1√ u

for EW,

4

 µ

c + 1

csech−1

√ u

for RLW,(3.4)

and the corresponding time duration is τ  = λ/ν. For any specified amplitude ratio

u, solitary waves of the RLW equation exhibit a different width for different wave

amplitudes, because the wave number k and width λ depend on both the constants µ

and c (one-third wave amplitude). In contrast, solitary waves of the EW equation have

a constant width for arbitrary wave amplitudes and speeds, because k and λ depend

only on µ. This is the special feature after which the EW equation was named. For thespecific case when u = 1/2 for the EW equation, the width at the one-half amplitude

level is λ = 4√ 

µ sech−1

1/√ 

2

= 4√ 

µ ln

1 +√ 

2

= 3.5255√ 

µ. If the width

of the solitary wave had been defined alternatively as λ = k−1 = 2√ 

µ, then λ would

correspond to another specific width of the solitary wave measured at the amplitude

ratio u = sech2(1/2) = 0.78645.

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Olver [22] has shown that solutions of the EW equation, like those of the RLW

equation, have only three conservation laws that can be written in the general form

T t +Xx = 0. These laws are the equivalents of the conservation of mass, momentum,

and energy in fluid mechanics. Olver showed that the three laws lead directly to threeso-called invariants of motion given by

C1 = +∞

−∞u dx , C2 =

 +∞

−∞

u2 + µux ux

dx , C3 =

 +∞

−∞u3 dx , (3.5)

provided that the integrals converge. These invariants of motion for the EW equation

need to be extended for this study. This is done by multiplying the EW equation,

ut  + uux − µuxx t  = 0, by 1, u, and u2 − 2µuxt , and then the resulting three

equations can each be expressed in the form T t 

+Xx

=0, which are summarized

as (u)t  +

12

u2 − µuxt 

x

= 0,

u2 + µux ux

t +

23

u3 − 2µuuxt 

x

= 0, andu3

t +

34

u4 − 3µut ut  − 3µu2uxt  + 3µ2uxt uxt 

x

= 0. These three conservation

lawscan now be integratedeasilywithrespectto x over a finite spatial domain [xL, xU ]instead of [−∞, +∞] to obtain the intermediate results

∂

∂t 

 xU 

xL

u dx + 1

2

u2

U  − u2L

= 0 ,

∂∂t 

 xU 

xL

u2 + µux ux

dx + 2

3

u3U  − u3L

= 0 , (3.6)

∂

∂t 

 xU 

xL

u3 dx + 3

4

u4

U  − u4L

= 0 ,

after simplifications. In these equations, uL = u(xL, t) and uU  = u(xU , t) are time-

invariant constantsat thedomainboundaries. In thesimplifications, the terms [uxt ]xU xL

,

[uuxt ]xU xL

, [ut ut ]xU xL

, [u2uxt ]xU xL

, and [uxt uxt ]xU xL

are zero at the boundaries because uL,

and uU  are constants thereat. Equation (3.6) can now be integrated with respect to t 

to yield

C1 = xU 

xL

u dx + 1

2

u2

U  − u2L

t ,

C2 = xU 

xL

u2 + µux ux

dx + 2

3

u3

U  − u3L

t , (3.7)

C3 = xU 

xL

u3 dx + 3

4

u4

U  − u4L

t .

These invariants of motion are generalizations of those given by Equation (3.5),

extended for the case of a finite length spatial domain when u(x,t) is constant but

not necessarily zero at the domain boundaries. The extra terms stem directly from

the convection of mass, momentum, and energy into and out of the lower and upper

boundaries of the spatial domain. These invariants of motion are equal to the initial

(t  = 0) mass, momentum, and energy inside the domain [xL, xU ]. Note that during

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numerical computations that provide solutions to the EW equation, C1, C2, and

C3 can be calculated after each successive time step over the entire spatial domain

xL ≤ x ≤ xU  that contains the wave motion, such that the conservation properties of 

the numerical algorithm can be monitored and thereby assessed.

3.3 Numerical Solution Procedure

A fairly complete description of the numerical solution procedure for the equal

width (EW) equation, ut 

+uux

−µuxx t 

=0, is given in this section. The method

consists in essence of numerically integrating this partial differential equation (PDE)forward in time to advance the solution u(x,t) at every node of a spatial grid, with

u(x,t) specified at each grid node at some initial time (e.g., t  = 0) and boundary

conditions applied at each time step to specify u(x,t) at the two edge nodes of the

grid. The solution of the EW equation on a uniform grid or nonuniform adaptive

grid requires discretizations of the spatial derivative terms ux and uxx t , and these dis-

cretizations can lead to a large set of implicit ordinary differential equations (ODEs),

one for each node. The resulting large set of stiff ODEs are integrated forward in time

by using an advanced ODE solver. This entire procedure is often called the method

of lines (MOL) for the sake of brevity. However, the numerical subprocedures in theMOL can vary substantially from one researcher to another; for example, the type

and order of the discretization, the type of the ODE solver, the use of a uniform or an

adaptive grid, and the method of interpolating u(x,t) and other data from a previous

to a new adaptive grid. Our numerical subprocedures and techniques are described

herein.

The EW equation, ut  + uux − µuxx t  = 0, is a time-dependent PDE in one space

dimension. To help describe the solution procedure more concisely, the EW equation

is written in functional notation as

f  (ut , uux , uxx t ) = 0, xL ≤ x ≤ xU , (3.8)

in which u is the dependent variable, x and t  are the independent variables, and xL

and xU correspond to the lower and upper limits or boundaries on x. As subscripts,

x and t  denote partial derivatives of the variable.

Initial conditions are specified before the solution procedure can commence. In

symbolic form they are stated as u0(x) ≡ u(x,t  = 0) for the finite domain xL ≤x

≤xU. The boundary conditions at xL and xU are required to determine a solution

either analytically or numerically. However, for most problems in which the solitary

or other wave propagation occurs well inside the boundaries the solution is negligible

at or outside the boundaries or interval [xL, xU] during the time span 0 ≤ t  ≤ t U  of 

consideration. Consequently, the boundary conditions at the lower and upper ends

of the interval [xL, xU ], given by u(xL, t) = uL and u(xU , t) = uU , are used, as

mentioned in the last section regarding the EW equation.

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A nonuniform spatial grid for the numerical solution procedure can be defined by

the vector x = [x1, x2, . . . , xi , . . . , xn]T , in which xi is the location of the ith node, n

is the total number of grid nodes, the superscript T  denotes the transpose of the vector,

and the lower and upper nodes x1 and xn correspond directly to the computationaldomain boundaries xL and xU. The distance or spacing between adjacent nodes can

be defined by xi = xi+1 − xi , for which i = 1, 2, . . . , n − 1. These xi are all

constant for a uniform grid, they vary in space for a nonuniform grid, and they also

vary in time for an adaptive grid.

The numerical solution is defined by the vector u = [u1, u2, . . . , ui , . . . , un]T , in

which ui is the numerical solution corresponding to grid node xi . This corresponds to

a discrete approximation of the PDEs in terms of space. At the initial time defined as

t 

=0, the solution vector u is initialized by using the initial data u0(x), which is also

discretized in space for the node locations x. The solution vector u is advanced intime as the numerical computations proceed, as will be described later. Note also that

∂ui /∂x and ∂ui /∂t  correspond to first-order derivatives at the ith node, and ∂ui /∂x

and ∂2ui /∂x2 are vectors of the first and second derivatives with respect to distance.

The spatial discretizations of the terms ux and uxx t  in the EW equation are ob-

tained by using finite-difference approximations on a nonuniform grid. These finite

differences can be expressed at node xi by

∂ ku

∂xk

xi

≈i+n

j =i−m

ci,j,kuj , k = 1, 2, . . . , (3.9)

in which normally m ≥ 0 and n ≥ 0, the number of grid points used for the derivative

is obviously m+n+1, which is called the stencil width, and ci,i−m,k, ci,i−m+1,k, . . . ,

ci,i+n,k are weights for the ithnodefor the kth derivative. These weights arespecific to

the type of derivative, being different for the general cases of forward finite differences

when m < n, centered finite differences when m = n, and backward finite differences

when m > n. The order of the first and second derivatives is given by m

+n when

k = 1 and 2, respectively, and higher order finite differences correspond directly to alarger stencil width.

Acronyms are used to denote various finite-difference schemes; for example,

cfd3p2o denotes a centered finite-difference scheme with m = n = 1, using a

three-grid-point stencil width of m + n + 1 = 3, and the first derivative is of or-

der m + n = 2. When centered finite differences cannot be used at and near the lower

and upper boundaries, appropriate forward and backward finite differences with the

same stencil width and order are used instead.

Two spatial discretization schemes are used in this study, and their influence on the

solution accuracy will be highlighted later. For the first scheme, labeled cfd5p4o, the

derivative ux is approximated by a five-point, fourth-order, centered finite difference

in space and stored as the vector ux , and the derivative uxx t  is approximated also

by a five-point, fourth-order, centered finite difference in space and stored as the

vector uxx t . For the second scheme, called cfd7p6o, the derivatives ux and uxx t  are

both approximated by a seven-point, sixth-order, centered finite difference in space.

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All spatial discretizations in this study were generated systematically by using the

versatile algorithm called WEIGHTS from Fornberg [8].

The discretization process for the EW equation, or Equation (3.8), on a spatial grid

produces a set of equations that can also be expressed in functional form as

f 1

u1, u2, . . . , un,

du1

dt ,

du2

dt , . . . ,

dun

dt , t 

= 0 ,

f 2

u1, u2, . . . , un,

du1

dt ,

du2

dt , . . . ,

dun

dt , t 

= 0 ,

......

...

f n u1, u2, . . . , un,du1

dt 

,du2

dt 

, . . . ,dun

dt 

, t  =0 ,

(3.10)

one equation for each grid node. This set of equations can be written concisely in

vector notation as

f 

u,

du

dt , t 

= 0 , (3.11)

and the initial conditions can be expressed likewise as

f 

u

t 0, du

dt 

t 0

, t 0

= 0 , (3.12)

in which u = [u1, u2, . . . , un], du/dt  = [du1/dt, du2/dt , . . . , dun/dt ] and f  =[f 1, f 2, . . . , f  n] for the grid x = [x1, x2, . . . , xn]. In the spatial discretization pro-

cess that resulted in Equations (3.10) and (3.11), the partial derivatives ux and uxx t 

of the EW equation have been expressed in terms of u and ut  at various grid nodes

(according to the type of discretization scheme). Because the only derivatives that

remain after the discretization are the partial derivatives ut 

|i

=∂ui /∂t  at various

nodes, these are then considered as, and replaced by, the total derivatives dui /dt ,which explains the sudden appearance of the total derivatives in Equation (3.10).

This discretization is called a semidiscretization because it occurs in space only and

not in time (i.e., the time derivatives remain).

The semidiscretization of the EW equation on a spatial grid, which involves the

space and the mixed space and time derivatives ux and uxx t , results in a fully implicit

set of ODEs in terms of u and its derivative du/dt , given by Equations (3.10) and

(3.11). This system of ODEs is fully implicit because the vector of derivatives du/dt 

is defined implicitly through the arguments of the vector function f . In other words,

the system of ODEs cannot be written in the explicit form du/dt  = f(u,t).

For some problems with the EW equation, the functions f 1, f 2, . . . , f  n in Equa-

tions (3.10) and (3.11) each contain one or more derivatives of u(x,t) with respect

to time, and all of the equations of the system are therefore differential equations

(DEs). For other problems, some of the functions f 1, f 2, . . . , f  n do not contain any

derivative terms and those that do not are algebraic equations (AEs). When the system

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given by Equations (3.10) and (3.11) contains a mixture of DEs and AEs, they are

called differential-algebraic equations (DAEs), or a system of differential-algebraic

equations. DAEs normally stem from the application of boundary conditions that are

algebraic in form, making f 1 and f n algebraic equations.The first general method for solving a system of DAEs like Equations (3.10) and

(3.11)was proposedoriginallybyGear [13]in1971. His basic ideawas toreplace each

derivative dui /dt  in Equation (3.11) by a backward differentiation formula (BDF) as

an approximation, so that the resulting system of nonlinear algebraic equations can

be solved for the vector uj  at time level t j , by using a suitable iterative method

such as a Newton iterative procedure. For example, consider the simple implicit

Euler formula uj  = uj −1 +duj 

dt  t j  − t j −1

, which can be rearranged to give the

first-order BDF as duj 

dt = uj  − uj −1

t j  − t j −1. Equation (3.11) can then be expressed as

f 

uj ,

uj  − uj −1

t j  − t j −1, t j 

= 0, which is a set of nonlinear algebraic equations that can

be solved for uj  at time level t j , by using a modified Newton iterative procedure such

as that given in the book by Ascher and Petzold [2]. However, a first-order BDF is

not sufficiently accurate for the time integration of the EW equation in this study.

Gear’s [13] more time-accurate integration procedure involves replacing the first-

order implicit Euler formula by the more accurate higher order implicit formula

uj  =q

=1

α uj − + β0

duj 

dt 

t j  − t j −1

, (3.13)

in which the integer q is the order of the BDF, and the real constants α (1 ≤ ≤ q)

and β0 have values that depend on the order q. The method of determining the

best set of values for these coefficients to optimize time integration accuracy and

ensure integration stability, and a tabulated set of the resulting values for the cases

of 1 ≤ q ≤ 6, are both given in the original paper by Gear [13]. These tables arerepeated in the literature and books by, for example, Brenan et al. [6] and Ascher

and Petzold [2]. Note that for the first-order case when q = 1, the coefficients have

values given by α1 = β0 = 1 for the implicit Euler formula. For the second-order

case when q = 2, the values of the coefficients are given by α1 = 4/3, α2 = 1/3,

and β0 = 2/3.

When the previous, more general formula is used to replace the vector of time

derivatives in Equation (3.11), one then obtains

f 

uj ,

1

β0(t j  − t j −1)

k=0

(−α) uj −, t j 

= 0 , (3.14)

in which now starts from zero and α0 = −1 is defined to include the uj  term.

This is a higher order time-accurate set of nonlinear algebraic equations that can

be solved for uj  at the j th time level by using a Newton iterative procedure. The

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BDF of Equation (3.13) is used to obtain Equation (3.14) because the solution values

for uj −1, uj −2, etc. from earlier time levels are known, whereas for centered and

forward finite-difference formulae the solution values of  uj +1, uj +2, etc. at future

time levels are still unknown. The BDF is also used because the implicit nature of uj occurring independently and as part of duj /dt  in Equation (3.13) provides important

stability properties for the time integration. The coefficients α (1 ≤ ≤ q) and β0

are determined in essence by a procedure that optimizes integration accuracy while

ensuring good integration stability.

The previous discussion about solving DAEs according to Gear’s original ideas and

Equations (3.13) and (3.14) implies that the coefficients α (1 ≤ ≤ q) and β0 are

all constants for a given order q of the BDF. This approach gives good solutions for

DAEs when the solutions are computed with equal-sized time steps, and also when

solutions for u have relatively smooth temporal variations, conditions for which the

original time-integration method was designed. However, for solutions that vary rel-

atively rapidly in time (i.e., high temporal gradients and curvatures in u), the time

steps should be reduced considerably and frequently to accurately capture any high

temporal gradient and curvature phenomena, and in such cases the method of fixed

coefficients can result in reduced solution stability. The remedy for this problem is

to incorporate variable coefficients that depend inherently on the variable time steps

which in turn dependon temporal solution gradients. Although themethod of variable

coefficients is the most stable implementation of the BDF methods, we do not rec-

ommend its implementation in the fullest form because of the disadvantage involving

computational inefficiency. The full approach requires numerous, computationally

laborious, Jacobian matrix evaluations or updates with time-step intervals of variable

size. See Brenan et al. [6] for more details.

The scheme for integrating Equation (3.14) forward in time, one time step after

another, should have the capability of using variable time steps to obtain accurate

solutions to the EW equation, for the reasons just mentioned. The scheme should

also have the capability of using variable orders of the BDF, for the following reasons.

The accuracy of the time integration can be improved by using a higher order BDF(e.g., q = 5 instead of 3). However, the time-integration accuracy will be degraded

temporarily during thefirst few time steps. During thefirst time step t 1 from j  = 1 to

2, the integration must start with the order q = 1 of the BDF to determine u2, because

only u1 is known. For the second time step t 2 from j  = 2 to 3, one can then use

q = 2, and so on and so forth, until q = 5 can be used at the fifth and subsequent

time steps. The time-integration accuracy is also temporarily degraded when adaptive

gridding is implemented at the end of any time step. This happens because the grid

nodes are suddenly rearranged in the adaptive grid process, the numerical results

from the previous grid are then interpolated onto the new grid, and most information

related to the previous grid and earlier time integration process becomes obsolete.

Hence, the time integration needs to be re-started with order q = 1, then q = 2, etc.

until q = 5. These temporary degradations in the time integration can be partially

overcome or minimized by using very small time steps when the order q is reduced

to less than its maximum value (5 in the previous example). These remarks illustrate

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that the integration method that will yield an accurate solution of the EW equation

must solve a large set of implicit algebraic equations (one for each grid node) on an

adaptive grid and include special features involving variable time stepping, variable

BDF order sequencing and variable BDF coefficients.The fully implicit DAE system given by Equation (3.11), which is approximated

by the fully implicit set of nonlinear algebraic equations given by Equation (3.14), is

integrated numerically in time in this study by using an advanced DAE solver called

DASSL from Petzold [24]. This versatile solver has special features for solving stiff 

DAEs. It can be used to time integrate either ODEs or DAEs, such that different prob-

lems governed by the EW equation which involve differential or algebraic boundary

conditions can be solved easily with minor computer-code modifications. This solver

can be used readily with either uniform or adaptive grids. The integration time steps

are changed automatically by the solver to capture high temporal gradient and cur-vature features of the solution and simultaneously maintain solution accuracy and

integration stability. A combination of fixed and variable coefficients for α and β0,

called the fixed leading-coefficient method by Ascher and Petzold [2], is used in

DASSL. This is a compromise between the less stable but computationally efficient

fixed coefficient approach and the more stable but computationally expensive variable

coefficient approach, which results in less integration stability but still retains good

computational efficiency by needing fewer Jacobian evaluationscompared to the fully

variable coefficient approach. The order q of the BDF is changed by the solver when

necessary, and q is varied from 1 to 5 by the DASSL solver.

The solver DASSL includes a subroutine called RES that computes the residual

of the system of equations resulting from the semidiscretization of the EW equation

(or other similar PDEs). The primary inputs to RES are the independent variable

t , dependent variable vector u, and derivative vector ut , so that the residual vector

having the form

R u, ut 

, t  =u

t  +uT 

·u

x −µ u

xx t 

(3.15)

can be computed, with ux and uxx t  both known from previous finite-difference ap-

proximations. The purpose of the solverDASSL is to compute the dependent variable

vector u by using a modified Newton method such that the residual vector R(u, ut , t)

approaches zero and Equations (3.11) and (3.14) are satisfied.

An adaptive grid is implemented in this study to facilitate the resolution of high

spatial gradients and curvatures to reduce truncation errors thereat in the solutions

of the EW equation. To help describe the adaptive grid scheme, consider the vector

of grid nodes x = [x1, x2, . . . , xn] with xi−1 < xi < xi+1 and having known

locations at the initial time level t 1 = 0 or some later time level t j . Consider the

corresponding solution u = [u1, u2, . . . , un] as known initially at time t 1 or just

computed at some later time level t j . The node locationscorresponding to thesolution

may not be optimal, and a method of moving these nodes to more optimal locations

is a requirement of an adaptive grid.

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The movement of grid nodes is based on the discrete function defined by

si

=  xi

x1

(α

+a1b1

+a2b2)γ  dx, 1

≤i

≤n ,

b1 = min

β,ux (x,t  j )

,

b2 = min

β,uxx (x,t  j )

,

(3.16)

and the values of si , one for each node at time level t j , are computed by the trapezoidal

rule. For this discrete function, s1 = 0, si−1 < si < si+1, and sn is the maximum

value. A continuous piece-wise linear function from si can be constructed when

required.

The parameters a1, a2, α, and β in Equation (3.16) are user-defined positive con-

stants for a particular solution to the EW equation. These parameters are normallyadjusted or tuned  for each EW solution (normally by solving part of a problem a

few times), such that the adaptive grid will help produce a good solution to the EW

equation for a specific problem, while using a reasonable number of grid nodes and

a moderate computational effort, which is a trial-and-error process that involves sub-

 jective assessment. The values of a1, a2, α, β, and γ  should be chosen to help

equidistribute the truncation error throughout the grid, that is the truncation errors in

regions of high gradients and curvatures with clustered nodes are roughly the same

as those in regions of low gradients and curvatures with sparsely spaced nodes.

The values of si in the adaptive grid process depend on the first and second space

derivatives, ux and uxx , which are approximated by finite differences (e.g., cfd5p4o

and cfd7p6o) using the subroutine WEIGHTS of Fornberg [8]. In some previous

studies, ux and uxx were obtained by differentiation of a cubic spline that was fitted

to the discrete data ui vs. xi . This cubic-spline approximation was used instead of 

finite-difference approximations to help provide more smoothly behaved derivatives,

which yield a more gradual or smoother variation in successive node spacings in the

adaptive grid. However, this minor advantage is not worth the extra computational

effort (i.e., solving a tridiagonal matrix system for the spline coefficients) during theprocedure of solving the EW equation.

FIGURE 3.3

Grid adaptation using an equidistribution principle.

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The previous locations of the grid nodes help define the piecewise linear curve

s(xi ) defined by Equation (3.16), which is depicted in Figure 3.3 for only n = 9

nodes for illustration purposes, in the form of a one-to-one mapping or projection

x → s. The new locationsof the nodes are obtained bydefining the equidistributed setS i = s1 + (sn −s1)(i −1)/(n−1), or more simply by S i = sn(i −1)/(n−1) because

s1 = 0. These S i are shown equidistributed along the vertical axis in Figure 3.3.

The inverse one-to-one mapping S  → x provides the new locations of the nodes,

as illustrated in Figure 3.3. The implementation of equal increments in the inverse

mapping procedure, by using theequidistribution constant S  = S i −S i−1 = sn/(n−1), with 1 ≤ i ≤ n, defines the essence of the equidistribution principle. Note that

during grid adaptations the first and last nodes x1 and xn automatically remain fixed

at the domain boundaries xL and xU .

Basic information that is relevant to understanding and specifying values for a1,a2, α, β, and γ  are now provided. The parameters a1 and a2 are generally set either to

zero or unity. For the combination a1 = 1 and a2 = 0, the adaptive grid movement is

based in essence on the magnitude of the solution gradient, that is |ux (x,t)|, whereas

the reverse combination of a1 = 0 and a2 = 1 bases the adaptive grid movement in

essence on the magnitude of the solution curvature that is typified by |uxx (x,t)|. In

this investigation, we normally set a1 = a2 = 1 such that the adaptive movement of 

the grid nodes is based on a combination of both the solution gradient and curvature.

The value of γ  has historically been set equal to 1/2. This most likely stems from

the earliest work in which the grid adaptation was based on the length of the curveu(x,t j ) vs. x, which is related to the solution gradient, so that the discrete function

was defined originally by si = xi

x1

 1 + u2

x (x,t  j ) dx. In more recent publications,

and also in this study, the values of  b1 and b2 in Equation (3.16) are not squared,

so retaining a value of γ  equal to 1/2 no longer seems rational. Nonetheless, we set

γ  = 1/2 in this study because the use of other more appropriate values has not been

explored.

For the following discussion about α and β, consider an important feature of the

equidistribution principle. Although S  = sn/(n − 1) exactly, it is also given ap-

proximately by

S  ≈ xnew

i+1

xnewi

(α + a1b1 + a2b2)γ  dx ≈ (α + a1b1 + a2b2)γ  xi (3.17)

in terms of the integrand and grid node spacing xi . When the discrete solution

for u(x,t j ) is constant or devoid of waves in some region or regions of space (e.g.,

near boundaries or between solitary waves), that is when ux

=0 and uxx

=0 in

such regions at the j th time level, then one can deduce from Equation (3.17) and the

definitions of b1 and b2 from Equation (3.16) that the node spacings are a maximum

for such conditions and given by xmax ≈ S/αγ . The parameter α, therefore,

plays a fundamental role in controlling the maximum node spacing.

The maximum node spacing can be specified by xmax = gmax(xU  − xL)/(n − 1)

for many problems, which defines this node spacing at a value of  gmax times the

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average node spacing, where gmax ≈ 3 to 8 is reasonable based on our experience.

Hence, we obtain

α ≈ S 

xmax

1/γ 

≈ 1

gmaxsn − s1

xU  − xL

1/γ 

, (3.18)

which shows in essence that αγ  is a fraction of the overall slope (sn − s1)/(xU  −xL) of the piecewise linear curve s(x), because gmax > 1. This result also shows

clearly that the parameter α is problem dependent by means of sn, which is the total

integral evaluated from x1 to xn in Equation (3.16). Of equal importance is that α

is independent of the total number of grid nodes (n). Hence, once the value of α is

known for a particular problem, it need not be changed when the same problem is

re-solved using a different number of nodes.One approach of specifying α for a new problem is to guess the value of sn and

estimate α ≈ [sn/gmax(xU  − xL)]1/γ . A better approach is to determine sn directly

by using the discretized initial conditions at time level t 1, which in many cases is

a good estimation for the entire problem. A finer tuning of the value of α requires

the study of numerical results of computations to determine if the final numerical

solution can be judged satisfactory. Changes to the value of α can also be invoked

to reduce numerical errors in the predicted invariants of motion of the conservation

laws, if these laws are known.

For the discussion about β, let this parameter be set initially to the maximummagnitude of  ux (x,t  j ) when a1 = 1 and a2 = 0, the maximum magnitude of 

uxx (x,t  j ) when a1 = 0 and a2 = 1, or the larger value of the maximum values

of |ux (xi , t j )| and |uxx (xi , t j )| when a1 = a2 = 1. If these maximum values are

not known before a problem is solved numerically, then a suitable value of β might

be guessed. When (α + a1b1 + a2b2)γ  is a maximum, then the node spacing is a

minimum, and one can deduce from Equation (3.17) and the definitions of b1 and b2

from Equation(3.16)that this minimum spacing isgiven by xmin ≈ S/ (α + κβ)γ ,

in which κ

=1 for the first two cases (a1

=0, a2

=1; a1

=1, a2

=0)and1

≤κ

≤2

for the last case (a1 = a2 = 1). In this last and more complex case, κ = 1 mightoccur when |ux (x,t  j )| and |uxx (x,t  j )| are a zero and a maximum at the same spatial

location, or conversely κ = 2 only if |ux (x,t  j )| and |uxx (x,t  j )| are both a maximum

at the same spatial location (an impossibility). More generally, the value of  κ is

somewhat larger than but near unity.

The result xmin ≈ S/ (α + κβ)γ  illustrates that |ux (x,t  j )|, |uxx (x,t  j )|, or β,

which is normally much larger than α, plays a strong role in controlling the minimum

node spacing. For example, when β is set to a value larger than both |ux (x,t  j )| and

|uxx (x,t  j )

|, then these magnitudes help directly in concentrating nodes in regions

of large solution gradients and curvatures, which produce one or more separated

regions of minimum node separation. When β is set to a value somewhat smaller

than the maximum values of both |ux (x,t  j )| and |uxx (x,t  j )|, then some gradient

and curvature values are said to be clipped , by means of the expressions for b1 and

b2 in Equation (3.16), however some nodes are still concentrated in regions of large

solution gradients and curvatures, but the concentration is somewhat less than the

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previous unclipped case for a larger value of β. If β is set to zero, then a uniform

grid will occur with an average node spacing (xU  − xL)/(n − 1). These comments

help illustrate that the so-called clipping parameter β plays a fundamental role in

controlling the minimum node spacing, such that solution gradients and curvaturesthat are large are well resolved by node clustering and those that are small are also

well resolved by means of sparsely spaced nodes. Consequently, the values of  α

and β control the node spacings which in turn help make the truncation error more

equidistributed throughout the grid.

The minimum node spacing can be specified by xmin = gmin(xU  − xL)/(n − 1)

for many problems, which sets the minimum node spacing at a value of gmin times the

average node spacing, where gmin ≈ 1/2 to 1/9 is reasonable based on our experience.

Hence, we obtain

β ≈ 1

κ

S 

xmin

1/γ 

− α

≈ 1

κ

S 

xmin

1/γ 

(3.19)

or

β ≈ 1

κ

1

g1/γ min

− 1

g1/γ max

sn − s1

xU  − xL

1/γ 

≈ 1

κ

1

gmin

sn − s1

xU  − xL

1/γ 

, (3.20)

which shows in essence that (κβ)γ  isa multiple of the overall slope (sn

−s1)/(xU 

−xL)

of the piecewise linear curve s(x), because gmin < 1. These results illustrate that theparameter β, like α, is problem dependent via sn but independent of the total number

of nodes.

The relationships for α and β can be combined to give xmax/xmin ≈(1 + κβ/α)γ  and κβ ≈ α

g

1/γ max/g

1/γ min

− 1

, provided that β is set sufficiently small

so that some clipping occurs. These results help illustrate the inter-relationships that

exist between the parameters α and β, xmax and xmin, and gmax and gmin. If we

take γ  = 1/2, gmax = 4, and gmin = 1/5, we can then estimate κβ ≈ 399α, which

illustrates that κβ/α 1 or κβ α.One approach of specifying β for a new problem is to start from a previously

guessed or calculated value of α and estimate β by means of Equation (3.20) or κβ ≈α

g1/γ max/g

1/γ 

min− 1

. On one hand, if solution gradients and curvatures are relatively

large, then the value of β will be less than the maximum values of |ux (xi , t j )| and

|uxx (xi , t j )|, clipping of the gradient and/or curvature will occur, and the maximum

and minimum node spacings will correspond closely to those expected from the

specification of the values of gmax and gmin. On the other hand, if solution gradients

andcurvatures arerelatively small, then the valueof β willbe larger than the maximum

values of |ux (xi , t j )| and |uxx (xi , t j )|, clipping of both the gradient and curvature will

not occur, and the maximum node spacing will not correspond closely to that expected

from the specification of the value of gmin. The nodes will be less concentrated than

expectedin regionsof higher gradientsandcurvatures, but a highernodeconcentration

will probably not be needed to obtain an accurate solution to the EW equation. A finer

tuning of the value of β, like the tuning of α, requires the study of some numerical

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solutions to determine if the final solution can be judged satisfactory. Changes to

the value of β may also be invoked to help reduce numerical errors in the predicted

invariants of motion from the conservation laws, if these laws are known.

The adaptive grid technique used in this study is based on the equidistributionprinciple described earlier. The nodes are held fixed during each time step. They are

moved only after a specified time interval denoted by t grid, which may include a

large number of very small time steps. This is called the static grid method because

the nodes are not moved simultaneously as the solution is computed, in contrast to the

dynamic grid method. The static adaptive node movement is illustrated in Figure 3.4,

where the grid adaptation is depicted after every four time steps. Although the grid

adaptation is done at the end of a time step, the node shifts are shown to occur over

an entire time step, but this is done for illustration purposes only.

FIGURE 3.4

Adaptive grid movement after every four time steps.

The grid adaptation is done after a reasonably short preset time increment thatmay include a few or a large number of time steps, depending somewhat on the

solution behavior. During this preset time increment t grid, the solution u(x,t) at the

grid nodes should not change by more than a few percent, or a solitary wave should

propagate only a short distance of a few nodes or less. In other words, the moving

parts of solutions with high gradients and curvatures should not outrun their region

of clustered nodes, if these high gradients and curvatures are to be properly resolved.

When the grid is adapted and updated, the solution from the previous grid needs

to be mapped onto the new grid. In this study this mapping or interpolation is done

by means of the quintic polynomial u(x,t j ) = 5k=0 bkxk between two successive

nodes xi and xi+1, and the polynomial coefficients bi are determined from our already

known values of ui , ux |i , and uxx |i at node xi and ui+1, ux |i+1, and uxx |i+1 at node

xi+1. This interpolation is fairly consistent, but not entirely consistent, with our five-

and seven-point finite-difference schemes cfd5p4o and cfd7p6o of fourth and sixth

order, respectively. The quintic polynomial u = 5k=0 bkxk can be written in the

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convenient prepackaged form

u = ui (1 − η)3

1 + 3η + 6η2

+ ui+1(1 − ξ )3

1 + 3ξ  + 6ξ 2

+ ux |i (1 − η)3η(1 + 3η)xi − ux |i+1 (1 − ξ )3ξ(1 + 3ξ)xi

+ 1

2uxx |i (1 − η)3η2x2

i + 1

2uxx |i+1 (1 − ξ )3ξ 2x2

i , (3.21)

so that all coefficients are expressed directly in terms of u and its first and second space

derivatives at adjacent nodes xi and xi+1. In this equation, the normalized distances

η = (x−xi )/xi and ξ  = 1−η, and xi = xi+1−xi . In previous research papers the

interpolation was done with cubic and quintic splines. The cubic spline interpolation

is simply insufficiently accurate for this study, as it was also for the previous studies

that used high-order finite-difference schemes. The quintic spline may be sufficientlyaccurate but one disadvantage is that it alters the first and second derivatives of u of 

the computed solution to make uxx x and uxxxx continuous. A further disadvantage

of cubic and quintic splines is the extra computational effort in solving tridiagonal

and pentadiagonal matrix systems for the spline coefficients, respectively, in contrast

to using a prepackaged interpolant like that given by Equation (3.21), for which the

derivatives ux and uxx are readily available.

When thetime integrationis restartedafter a gridadaptation, the solverDASSL isre-

initialized by using the parameterINFO, which is set to a value of zero sothat theorder

q of the BDF restarts from unity, because the previous obsolete solution values andJacobian entries correspond to the previous rather than the new node locations. This

proceduregave good results, so themore sophisticated, accurate, andcomputationally

intensive method of remapping u(x,t) and all related information from the previous

q time levels and previous grid to the new grid were not implemented.

Numerical solutions for the EW equation always begin on a nonuniform grid. The

initial conditions at time t  = 0 are initially discretized on a uniform grid, then this

uniform grid is adapted to a nonuniform grid by using these initial conditions. The

initial conditions are then re-interpolated onto the nonuniform grid. This grid should

be adapted once more to a new nonuniform grid, which will differ somewhat from theprevious one because the mapping of x → s uses approximate trapezoidal integration

and the mapping S  → x uses a piecewise linear curve. The initial adaptive grid is

then prepared for the numerical computations.

One important feature of the previously described adaptive grid technique is that

this scheme is independent of, or uncoupled from, the PDE discretization and time

integration procedures, because the grid adaptation is done between time steps when

the discretization and integration procedures are halted. Hence, the adaptive grid

algorithm is problem independent, and it can therefore be coded once for all problems.

In this study, the adaptive grid subroutine called AGE is used, which was obtained

from Saucez et al. [28]. Similar algorithms such as those by White [31], Sanz-Serna

and Christie [27], and Revilla [25] can be implemented easily from the ideas presented

in their papers.

It is important to realize that the use of more nodes (which results in a larger set

of DAEs) with a larger variation in the node spacings, increases the computational

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effort and related computer run times. Furthermore, closely spaced nodes make the

set of DAEs computationally stiff  and the problem solution by means of the solver

DASSL requires an increased number of smaller time steps to produce a full solution,

which leads to longer central processor run times. See the books by Brenan et al. [6]and Schiesser [30] for a detailed definition and explanation of the stiffness of DAEs,

which is directly related to the separation of the eigenvalues of the Jacobian matrix

of the DAE system, which in turn is adversely affected by large variations in node

spacings. Because node spacings vary widely for problems solved numerically in this

study, the double precision version of the stiff DAE solver DASSL is used to integrate

Equation (3.14). Also, an approximate Jacobian is computed internally by this solver

using finite differences. Small absolute and relative tolerances on local time steps are

imposed by setting the values of ATOL and RTOL in DASSL to the small value 10−8,

and this suppresses the time integration errors to negligibly small values.The banded structure of the Jacobian matrix is an important consideration for the

numerical computations done in this study. The two spatial discretization schemes

cfd5p4o and cfd7p6o used in the fully implicit DAE time integration by the DASSL

solver lead to septagonal and endectagonal banded Jacobian matrices, respectively.

Hence, for cfd5p4o and cfd7p6o, the half-bandwidths ML and MU in the DASSL

solver are each set to 3 and 5, respectively, resulting in the Jacobian bandwidth

defined by ML + MU + 1 to have values of 7 and 11, respectively. Note that higher

order discretizations produce Jacobian matrices with larger bandwidths, which in

turn increase the computational effort to produce the numerical solution. However, alower order finite-difference scheme coupled to a grid with lots of nodes will yield a

reasonably accurate solution, but the computations will be inefficient.

3.4 Numerical Results and Discussion

3.4.1 Single Solitary Waves

The EW equation, ut  + uux − µuxx t  = 0, is solved in this section to predict

the motion of a single solitary wave in space and time. This problem is solved for

the case of the parameter µ = 1/16, by using the MOL whose subprocedures were

outlined previously. Many solutions are obtained using uniform and adaptive grids

with different numbers of grid nodes, and also using two different finite-difference

schemes. The exact solution for this problem is known and given in general by

Equation (3.2) as u(x,t)

=3 c sech2[k (x

−x0

−νt )]. The wave number is k

=1/√ 4µ = 2, the wave speed is ν = c = 1/10, the peak amplitude is upeak  = 3c =3/10, and the wave is centered at x0 = 20 at t  = 0. Many exact and numerical results

are compared for this benchmark problem to assess various parts of the solution

procedure.

The initial conditions for the numerical computations are specified by using the

exact solution at time t  = t 1 = 0 as u(x, 0) = 3 c sech2[k (x − x0)] on the interval

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[xL = 0, xU  = 45]. The numerical solution is computed for times varying from

t  = 0 to 60 with various spatial discretization schemes on uniform and adaptive grids

with the number of nodes varying from 51 to 401. During the time interval 0 to 60,

the solitary wave is always far from the grid boundaries, so the Dirichlet boundaryconditions u(xL, t) = uL = 0 and u(xU , t) = uU  = 0 are applied because they are

sufficiently accurate for the current problem.

The exact solution is given in Figure 3.5 for the spatial and temporal intervals

0 ≤ x ≤ 45 and 0 ≤ t  ≤ 60, respectively. This problem becomes challenging to

solve numerically on uniform and adaptive grids when the number of nodes is reduced

sufficiently such that the solution gradients and curvatures become difficult to resolve

accurately.

 

FIGURE 3.5

Solitary wave motion calculated with the exact solution.

The MOL solution of the EW equation for our problem with µ = 1/16 givesnumerical results for the motion of a single solitary wave with a nearly constant

speed, peak amplitude, and shape. The discrete solution u(xi , t j ) is normally in

close agreement with the exact solution shown in Figure 3.5, if the grid nodes are

suf ficiently numerous, such that the exact and numerical solutions overlap and are

not readily distinguishable. Hence, the differences between the exact and numerical

solutions, uexacti − unum

i , as a function of distance at a given time should be used, and

these errors are shown in Figure 3.6. Two piecewise linear solutions are displayed

for the cases of n

=101 grid nodes, time t 

=60, an adaptive grid with parameters

a1 = a2 = 1, α = 0.01, β > 2.5, grid adaptation implemented after the presettime interval t grid = 1, and for the finite-difference schemes cfd5p4o and cfd7p6o

defined earlier.

Both of the discretization schemes in the MOL give fairly accurate and acceptable

solutions, with errors relative to the peak amplitude upeak  = 3/10 that are less than

2%. However, the finite-difference scheme cfd7p6o with the larger stencil and higher

order gives more accurate results, as might be expected, with errors less than 0.3% (vs.

2% for the smaller stencil). If more grid nodes are used, the errors will then decrease.

However, these numerical solutions were presented mainly to show that the scheme

cfd7p6o with the wider stencil gives more accurate results and is a better choice for

solving such problems, and increasing the number of nodes does not change this

conclusion. Other results given later will further justify this remark. Our numerical

solutions are also presented to show that these results are more accurate than those

published earlier by Gardner et al. [10] for a much easier problem with µ = 1, for

which the wave is much smoother spatially in that it is four times wider. The errors

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FIGURE 3.6

Solution errors for cfd5p4o and cfd7p6o.

for the case of our adaptive grid with 101 nodes are three times smaller than those of 

Gardner et al. [10] for the case of a uniform grid with 1001 nodes.

For our problem with µ = 1/16 and k = 2 for a single solitary wave, any value

of β > 2.437 yields the same results for the following reasons. The maximum value

of |ux (x,t)| is 0.4619 at x = x0 ± 0.3292 (at which u = 0.20 and uxx = 0), where

the bar over the last digit of a number implies that this digit continues indefinitely.

The maximum value of |uxx (x,t)| is 2.40 at x = x0 (where u = 0.30 and ux = 0).The maximum value of |ux (x,t)| + |uxx (x,t)| is 2.437 at x = x0 ± 0.03111. Values

of β set larger than this last maximum do not invoke any clipping by means of the

expressions for b1 and b2 given by Equation (3.16). Clipping will occur for lower

β values, that is when β < 2.437, but this clipping is unnecessary because the grid

nodes are clustered appropriately in regions of high gradients and curvatures, and

they are not overly crowded anywhere in the numerical solutions in this section. In

other words, the minimum node spacing was not overly small in regions of maximum

gradients and curvatures such that a stiff set of DAEs became dif ficult to solve. For

the current problem, gmax ≈ 1.452 and gmin ≈ 1/10.61, meaning that the maximumand minimum node spacings are about 3/2 times larger and 11 times smaller than the

average node spacing.

The MOL solutions of the EWequationfor the sameproblem using a coarseuniform

grid and an adaptive grid with only 101 nodes are shown in Figures 3.7 and 3.8. Each

set of solutions for u(xi , t j ) pertains to time levels t j  of 0, 20, 40, and 60, the solutions

are given on the small spatial interval [15, 33] of the whole interval [0, 45] for clarity,

and other problem and computational data are included as an inset in these figures.

The node locations for the solutions at various times are shown in the lower part of 

each figure. For both sets of solutions the discrete data are connected by straight

lines, which is a low-order solution reconstruction or interpolation. For the first set of 

results in Figure 3.7 the discrete data is also connected by a smooth curve using the

quintic interpolation given by Equation (3.21). The piecewise linear solutions appear

rather kinky for the case of the uniform grid and quite smooth for the nonuniform

grid, illustrating that the adaptive grid that concentrates nodes in regions of higher

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gradients and curvatures helps resolve or capture such steep solution gradients and

curvatures more effectively. This is the primary reason for comparing these two sets

of solutions, and other forthcoming results will also show that the adaptive grid yields

superior solutions.

FIGURE 3.7

Single solitary wave solutions using a uniform grid.

FIGURE 3.8

Single solitary wave solutions using an adaptive grid.

The MOL solution for the solitary wave on the uniform grid with only 101 nodes

is quite inaccurate based on the results in Figures 3.7 and 3.8. For example, the peak 

amplitude of the wave on the uniform grid is less than the exact value of 3/10, the

wave is propagating too slowly in that the center of this wave has moved to only

x ≈ 25 at time t  = 60 instead of the exact value x = 26, and the dip below zero

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in the solution between x = 18 to 21 should not occur. Solutions on a uniform grid

with more nodes would, of course, be more accurate.

Although the piecewise linear reconstruction of the solutions in Figures 3.7 and 3.8

is a convenient stratagem to illustrate the benefits of using an adaptive grid in compar-ison to a uniform grid, this stratagem is unfair. The finite-difference scheme cfd7p6o,

or the quintic equation given by Equation (3.21) for interpolating solutions from a

previous to new adaptive grids, should be used instead to reconstruct a smooth solu-

tion for u(x,t) between nodes. When this is done the solutions for both cases are no

longer piecewise linear with kinkiness, especially apparent for the case of the coarse

uniform grid, but they will be smooth as illustrated by the interpolated results shown

in Figure 3.7. These interpolated sets of solutions represent much better the solutions

obtained by the MOL. Nevertheless, the solution computed on the adaptive grid is

still more accurate than that on the uniform grid, as we can see qualitatively from the

results in Figures 3.7 and 3.8. However, the accuracy cannot be easily determined

quantitatively from the results in these figures, so other results are now introduced.

The peak amplitude and its trajectory in space and time are given by the exact

solution as uexactpeak 

= 3c = 3/10 and xexactpeak 

= x0 + νt  = 20 + t/10, respectively,

because ν = c = 1/10 and x0 = 20 for our problem. The corresponding values

of unumpeak and xnum

peak  from the MOL solution can be obtained by searching through the

discrete data u(xi , t j ) at time level t j  = 60 for the maximum value and recording its

corresponding node location. Such simple results for the peak amplitude and locationare simply judged as being too crude for this investigation. Instead, an interpolation

of u(xi , t j ) between grid nodes is done by using Equation (3.21), and the maximum

(peak) amplitude and its spatial location are obtained by using a well-known iterative

procedure due to Brent [7]. These numerical results are presented in Table 3.1 for

the case of µ = 1/16 and the cases of an adaptive grid (a1 = a2 = 1, α = 0.01,

β > 2.5), two finite-difference schemes cfd5p4o and cfd7p6o, and different numbers

of nodes n equal to 51, 101, 201, and 401.

Table 3.1 Peak Amplitude and Its Location

unumpeak  xnum

peak 

n cfd5p4o cfd7p6o cfd5p4o cfd7p6o

51 0.28452 0.29824 25.8310 26.0129

101 0.29832 0.30005 25.9816 25.9968

201 0.29885 0.30001 25.9982 25.9998

401 0.29999 0.30000 25.9998 26.0000

For µ = 1/16, k = 2, t  = 60; adaptive grid with

a1 = a2 = 1, α = 0.01, β > 2.5, t grid = 1.

The corresponding numerical errors in the peak amplitude and its location are given

in Table 3.2 f or the cases of uniform and adaptive grids, and for the number of nodes

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varying from 51 to 401. These relative errors are defined by

eamp = 1 −max

xunum

peak (x,t)

maxx

uexactpeak (x,t) and ephase = 1 −

xnumpeak (x,t)

xexactpeak (x,t) , (3.22)

respectively, and the latter is the well-known phase error. The data in Tables 3.1

and 3.2 show, as one might expect, that the predicted peak amplitude and its location

are more accurate for finite-difference scheme cfd7p6o than cfd5p4o, and also when

a larger number of nodes is incorporated into the grid.

Table 3.2 Errors in Peak Amplitude and Its Locationeamp ephase

n cfd5p4o cfd7p6o cfd5p4o cfd7p6o

51 0.0472 0.00586 0.00650 −0.000496

101 0.00559 −0.000161 0.000707 0.000125

201 0.000497 −0.0000207 0.0000679 0.00000777

401 0.0000385 −0.00000107 0.00000579 0.000000547

For µ = 1/16, k = 2, t  = 60; adaptive grid with

a1

=a2

=1, α

=0.01, β > 2.5, t grid

=1.

Additional results for numerical errors in the peak amplitude and its location are

summarized in Table 3.3 for the cases of uniform and adaptive grids with 101 nodes,

and at times varying from t  = 0 to 60. The relative errors in amplitude and phase for

the case of the uniform grid are rather large, because 101 nodes are simply insuf ficient

to obtain a good MOL solution. However, these errors are considerably less and quite

acceptable for the case of the adaptive grid. Such tabulated results illustrate clearly

the advantages of using an adaptive grid over a uniform grid in the MOL for the case

of the same number of nodes.

Table 3.3 Relative Errors of Peak Amplitude and Phase

eamp ephase

Time Uniform Adaptive Uniform Adaptive

0 0.0680 0.000000169 0.00193 −0.000000105

10 0.0823 0.0000697 0.0137 −0.00000990

20 0.0870

−0.000174 0.0198

−0.00000738

30 0.0890 −0.000243 0.0249 0.000032040 0.0905 −0.000209 0.0293 0.0000671

50 0.0915 −0.000196 0.0330 0.0000970

60 0.0919 −0.000161 0.0362 0.000125

For µ = 1/16, k = 2, n = 101; adaptive grid with

a1 = a2 = 1, α = 0.01, β > 2.5, t grid = 1.

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The constancy of the three invariants of motion given by Equation (3.7) were

monitored during the MOL computations, and some of these results are presented

after our analytical and numerical methods of evaluating C1, C2, and C3 have been

explained. The integrals for the three invariants of motion given by Equation (3.7)can be evaluated analytically for our problem involving a single solitary wave. The

exact results are

Cexact1 = 6 c

k= 12c

√ µ , Cexact

2 = 725

c2

k= 

1445

c2√ µ ,

Cexact3 = 

1445

c3

k= 

2885

c3√ µ , (3.23)

at t  = 0 and for uL = 0 and uU  = 0, leading to Cexact1 = 3/10 = 0.30,

Cexact

2 =9/125

=0.0720 and Cexact

3 =9/125

=0.01440 for our problem with

µ = 1/16 and c = 3/10. In the numerical calculations the values of Cnum1 , Cnum

2 , and

Cnum3 are evaluated in the following manner. The quintic interpolant given by Equa-

tion (3.21) is used to interpolate u(x,t) between nodes. For Cnum1 =  xU 

xLu(x,t j ) dx,

this interpolant is first integrated for a general grid interval, and the results for all inter-

vals are then summed to obtain Cnum1 . For Cnum

2 =  xU 

xL

u2(x,t  j ) + µu2

x (x,t  j ) 

dx,

the interpolant is squared, the differentiated interpolant is squared, the first part is

added to µ times the second part, these results are then integrated for a general

grid interval, and the results for all intervals can then be summed to get Cnum2 . For

C

num

3 =  xU 

xL u

3

(x,t  j ) dx the interpolant is first cubed, integrated, etc. The integra-tions for a general grid interval can be done fairly easily by using well-known software

such as Mathematica or Maple.

Table 3.4 Errors in Invariants of Motion for cfd5p4o and

cfd7p6o

Cnum1 Cnum

2 Cnum3

n cfd5p4o cfd7p6o cfd5p4o cfd7p6o cfd5p4o cfd7p6o

51 0.28452 0.30059 0.066297 0.071333 0.012701 0.014197101 0.29821 0.29962 0.071280 0.071982 0.014184 0.014394

201 0.29983 0.29996 0.071934 0.072000 0.014380 0.014400

401 0.29999 0.30000 0.071995 0.072000 0.014440 0.014400

For µ = 1/16, k = 2, t  = 60; adaptive grid with

a1 = a2 = 1, α = 0.01, β > 2.5, t grid = 1.

The numerically computed values of the three invariants Cnum1 , Cnum

2 , and Cnum3

are given in Table 3.4 for the cases of the different finite-difference schemes cfd5p4o

and cfd7p6o and with grid nodes varying from 51 to 401. The corresponding relative

errors are summarized in Table 3.5 for the same conditions. These two sets of results

clearly show that the invariants of motion are better preserved in the MOL solutions,

or the errors are smaller, for scheme cfd7p6o in contrast to cfd6p5o, and also when

the number of nodes is increased. An additional set of the relative errors for the

invariants of motion are given in Table 3.6 for the cases of uniform and adaptive grids

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Table 3.5  Relative Errors in Invariants of Motion for cfd5p4o and cfd7p6o

Cexact1 − Cnum

1

Cexact

1

Cexact2 − Cnum

2

Cexact

2

Cexact3 − Cnum

3

Cexact

3n cfd5p4o cfd7p6o cfd5p4o cfd7p6o cfd5p4o cfd7p6o

51 0.0516 −0.00196 0.0792 0.00926 0.118 0.0141

101 0.00598 0.00126 0.0100 0.000255 0.0150 0.000384

201 0.000566 0.000140 0.000912 −0.00000336 0.00137 −0.00000503

401 0.0000472 0.0000120 0.0000722 0.000000143 0.000109 0.000000215

For µ = 1/16, k = 2, t  = 60; adaptive grid with

a1 = a2 = 1, α = 0.01, β > 2.5, t grid = 1.

at various times from 0 to 60. These results clearly show, as expected, that the use

of an adaptive grid in the MOL computations is superior to the uniform grid for the

case of the same number of grid nodes.

Table 3.6  Relative Errors in Invariants of Motion for Uniform and Adaptive

Grids

Cexact

1 −Cnum

1Cexact

1

Cexact

2 −Cnum

2Cexact

2

Cexact

3 −Cnum

3Cexact

3

Time Uniform Adaptive Uniform Adaptive Uniform Adaptive

0 0.000712 0.0000509 0.0251 −0.000000276 0.0432 −0.000000125

10 −0.0253 0.000297 0.0345 0.0000317 0.0593 0.0000481

20 −0.0658 0.000489 0.0350 0.0000762 0.0611 0.0001152

30 −0.0772 0.000684 0.0354 0.000119 0.0621 0.000181

40 −0.0789 0.000872 0.0357 0.000166 0.0628 0.000250

50 −0.0792 0.00106 0.0358 0.000207 0.0633 0.000313

60 −0.0793 0.00126 0.0359 0.000255 0.0635 0.000384For µ = 1/16, k = 2, n = 101; adaptive grid with

a1 = a2 = 1, α = 0.01, β > 2.5, t grid = 1.

One might think that the relative errors in the three invariants of motion at time

t  = 0 in Table 3.6 should be identically zero, because at t  = 0 the solution for the

solitary wave motion is started with initial conditions obtained directly from the exact

solution, and the MOL solution procedure has yet to begin. However, the integration

scheme using discrete data from the grid to evaluate Cnum

1

, Cnum

2

, and Cnum

3

and using

the interpolant given by Equation (3.21) is approximate. In the present problem,

these spatial integration errors associated with evaluating the invariants of motion at

a given time are one to three orders of magnitude smaller than the time integration

errors associated with the solution procedure by the MOL. Hence, we believe that

the values of Cnum1 , Cnum

2 , and Cnum3 given in Table 3.4 and the relative errors given

in Tables 3.5 and 3.6 primarily reflect the errors due to the MOL solution. In other

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words, our reported errors for the MOL procedure are not dominated by the spatial

integration errors from determining the invariants of motion. If we had implemented

the spatial integrations to obtain the invariants of motion by using the trapezoidal

rule, Simpson’s rule, or a cubic spline, then the results would have been dominatedby spatial integration errors. Most previous studies incorporate a low-order spatial

integration scheme (e.g., Simpson’s rule or a cubic spline) to evalue the invariants

of motion, and their reported values of Cnum1 , Cnum

2 , and Cnum3 and their associated

relative errors are inaccurate, being contaminated by the spatial integration error.

Error norms have also been computed for the current problem because the exact

solution is known. We use the norm L2 = uexact − unum2, which is defined by

L2= 1

xU  − xL xU 

xLuexact

−unum2

dx1/2

(3.24)

≈

1

xU  − xL

n−1i=1

1

2(xi+1 − xi )

uexact

i − unumi

2 + uexact

i+1 − unumi+1

21/2

,

for the continuous and discrete error functions, respectively. The former is integrated

by the trapezoidal rule to get the latter, in which uexacti and unum

i are the exact and

numerical solutions at the ith node xi . We also use the norm L∞ = uexact−unum∞,

defined as

L∞ = maxuexact − unum

≈ maxi

uexacti − unum

i

(3.25)

for the continuous and discrete error functions, respectively.

Results for these error norms for the cases of different finite-difference schemes

and varying times from t  = 0 to 60, and for uniform and adaptive grids with different

numbers of grid nodes, are summarized in Tables 3.7 and 3.8. These tabulated data

illustrate once again that the MOL using the larger stencil associated with cfd7p6o

and an adaptive grid gives more accurate solutions than when the MOL uses a smallerstencil associated with cfd6p5o and a uniform grid with the same number of nodes.

Table 3.7 L2 and L∞ Errors for cfd5p4o and

cfd7p6o

L2 L∞n cfd5p4o cfd7p6o cfd5p4o cfd7p6o

51 0.00658 0.000636 0.0636 0.00439

101 0.000758 0.000139 0.00615 0.00103201 0.0000725 0.00000861 0.000587 0.0000621

401 0.0000062 0.00000060 0.000050 0.0000043

For µ = 1/16, k = 2, t  = 60; adaptive grid with

a1 = a2 = 1, α = 0.01, β > 2.5, t grid = 1.

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Table 3.8  L2 and L∞ Errors for Uniform and

Adaptive Grids

L2 L∞

Time Uniform Adaptive Uniform Adaptive

0 0.0 0.0 0.0 0.0

10 0.00810 0.0000195 0.0487 0.000121

20 0.0138 0.0000494 0.0889 0.000166

30 0.0185 0.0000494 0.128 0.000321

40 0.0229 0.0000764 0.159 0.000542

50 0.0273 0.000106 0.183 0.000774

60 0.0316 0.000139 0.206 0.001032

For µ=

1/16, k=

2, n=

101; adaptive grid with

a1 = a2 = 1, α = 0.01, β > 2.5, t grid = 1.

Some interesting information on the time integration of the EW equation with

µ = 1/16 for the motion of the solitary wave by the DAE solver called DASSL in the

MOL is presented in Table 3.9. The total number of time steps to solve the problem

by the MOL from time t  = 0 to 60 ranges from 2894 to 2945 for the two finite-

difference schemes cfd5p4o and cfd7p6o and grid nodes varying from 51 to 401. The

size of each time step is selected by the solver to ensure that the time integration isdone accurately, and these steps are for the most part independent of the number of 

nodes. These time steps are fairly constant at about 0.02 because the solitary wave

propagates with a constant shape and speed, so the time integration is of the same

degree of dif ficulty for each time step.

Table 3.9 Time Steps, Function and Jacobian Calculations, and CPU Times

Total Number Total Number of Total Number of CPU Time

of Time Steps Function Evaluations Jacobian Calculations (s)

n cfd5p4o cfd7p6o cfd5p4o cfd7p6o cfd5p4o cfd7p6o cfd5p4o cfd7p6o

51 2894 2930 4339 4376 1202 1202 62 85

101 2942 2945 4388 4391 1202 1202 124 169

201 2945 2945 4391 4390 1202 1202 246 341

401 2944 2944 4388 4330 1202 1202 491 668

For µ = 1/16, k = 2, t  = 60; adaptive grid with

a1 = a2 = 1, α = 0.01, β > 2.5, t grid = 1.

The number of function evaluations used in the solver DASSL is related to the

number of times the DAEs are used in the full solution procedure, whereas the number

of Jacobian calculations are the number of times the entries of the Jacobian matrix

are calculated or updated. The number of function and Jacobian calculations are also

related to the time integration and not to the number of nodes, which explains why

they are approximately constant for schemes cfd5p4o and cfd7p6o and also for the

grid nodes varying from 51 to 401.

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The central processor (CPU) time needed by the computer to determine a complete

MOL solution to the current problem is directly proportional to the number of grid

nodes and size of the grid-point stencil of the finite-difference scheme. The CPU times

given in Table 3.9 are those for a Pentium III computer with a 500-MHz processorand LINUX operating system. From the tabulated data one can see that the CPU

time almost exactly doubles as the number of nodes doubles for both cases of finite-

difference schemes cfd5p4o and cfd7p6o. The CPU time for scheme cfd7p6o is

about 37% larger than that for scheme cfd5p4o, for each case with the same number

of nodes. This occurs mainly because scheme cfd7p4o has a seven-point stencil and

an eleven-band Jacobian matrix compared to the five-point stencil and seven-band

Jacobian matrix for cfd5p4o, which are respectively 40% and 57% larger for scheme

cfd7p6o, making computations for the larger stencil cfd7p6o more tedious and longer.

Solutions to the current solitary wave problem by the MOL using the finite-difference scheme cfd7p6o as compared to cfd5p4o are obtained more ef ficiently

by about 30%, in terms of the CPU time for numerical results obtained with the same

accuracy. From Tables 3.2, 3.5, and 3.7 we observe that solutions for scheme cfd5p4o

are about the same accuracy as those for scheme cfd7p6o when the number of grid

nodes for scheme cfd7p6o are one half of those for scheme cfd5p4o. Hence, from

Table 3.9 the CPU times of 85, 169, and 341 for 51, 101, and 201 grid nodes for

scheme cfd7p6o should be compared directly to the CPU times of 124, 246, and 491

for 101, 201, and 401 grid nodes for scheme cfd5p4o. This observation leads to the

conclusion that the CPU times are roughly 30% smaller for MOL solutions of the

same accuracy with scheme cfd7po vs. scheme cfd5po. Hence, the additional com-

puter programming effort in implementing a higher order finite-difference scheme

(larger stencil width) provides a payoff in terms of a modest reduction in the CPU

time for a given accuracy.

3.4.2 Inelastic Interaction of Solitary Waves

The interaction of two solitary waves, such as two waves that collide or one wavethat overtakes a slowerwave, arefascinatingfrom thepointofview that thetransmitted

waves may retain much of their original shapes and speeds after the interaction and

additional waves can be generated by the interaction process, because the interaction

is inelastic or uncleanfor the EW equation, as noted in the introduction. Such inelastic

interactions are also interesting from the point of view of the capability of the MOL to

obtain good solutions, because the predictions can be dif ficult in terms of accurately

capturing the wave interaction process and generation of small secondary waves.

Solutions to the EW equation, ut 

+uux

−µuxx t 

=0, with µ

=1 by the MOL

are presented for three different problems in this section. Two of these problemsinvolve the collision of two waves and the other involves the overtaking of one wave

by another. None of these problems have known analytical solutions to provide a

guide to, or a check on, the numerical results that are presented in this section.

The first problem for the EW equation with µ = 1 involves the collision of two

waves, a problem studied originally for the RLW equation by Santarelli [26]. One

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wave is specified initially at time t  = 0 by the solitary wave solution given by

Equation (3.2) as 3c1sech2 [k1 (x − x01)] with c1 = ν1 = 1.70, k1 = 0.50, and

x01 = 35, and the second wave is also specified initially by the solitary wave solution

as 3c2sech2

[k2 (x − x02)] with c2 = ν2 = −3.40, k2 = 0.50, and x02 = 65. Thesetwo waveforms are simply added together and used as initial conditions at t  = 0.

The MOL solution is obtained by using the finite-difference scheme cfd7p6o on an

adaptive grid with 201 nodes over the spatial interval [0, 80]. Solutions are computed

for the time interval [0, 16], and grid adaptation is done at regular time intervals of 

t grid = 2. The other adaptive grid parameters are a1 = a2 = 1, α = 0.001,

and β = 0.1. This value of β causes a significant degree of clipping because the

maximum value of |ux | + |uxx | ranges from a value of 6 for the initial solitary waves

to 40 during their interaction.

A time-distance diagram of the numerical computations is depicted in Figure 3.9.The rightward traveling wave with the positive amplitude (3c1 = 5.10) and speed

(ν1 = c1 = 1.70) and the leftward propagating wave with the negative amplitude

(3c2 = −10.20) and speed (ν2 = c2 = −3.40) eventually collide, and after their

interaction they become transmitted waves with shapes and speeds that appear sim-

ilar to their original ones. A closer examination shows that their peak amplitudes

and speeds are about 2% smaller. During the collision these two interacting waves

experience slight phase shifts, which are not readily seen in Figure 3.9. The collision

process also produces a small stationary disturbance between the transmitted waves,

which is easily seen. Santarelli [26] solved this problem first for the RLW equationand resolved the small disturbance, which he called a pair of daughter waves. His

work was later confirmed by Lewis and Tjon [20] and Gardner and Gardner [12].

FIGURE 3.9

Collision of two solitary waves leaving a stationary disturbance.

Two spatial distributions at times t  = 0 and 14 are shown more clearly in Fig-

ure 3.10, where the shape of the small stationary disturbance is enlarged for clarity

(the amplitude is magnified by a factor of 15). The distributions of grid nodes at the

two different times are also shown for interest in the lower part of the figure, to illus-

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trate the node clustering in solution regions of larger gradients and curvatures. The

solution by the MOL with only 201 nodes appears to provide a good solution to this

problem, from the viewpoint that the grid adaptation seems to well resolve solution

regions of large gradients and curvatures, including the transition through the smallstationary disturbance. For interest, the maximum and minimum node spacings for

this problem are about 5 and 1/2 times the average node spacing, respectively.

 

FIGURE 3.10

Collision of two solitary waves and details of the stationary disturbance.

The trajectories of the peak amplitudes of the two waves before, during, and after

the collision are plotted as solid lines in Figure 3.11. The shifts in these trajectories

from the extrapolated paths given by the dashed lines are shown clearly, whereas these

phase shifts were not obvious in Figure 3.9, even though these shifts are substantial

at

−1.43 and 2.64 for the positive and negative amplitude waves, respectively.

FIGURE 3.11

Collision of two solitary waves depicting their phase shifts.

The invariants of motion for spatially localized solitary waves and initial data

are given in general by Equation (3.7), and for a single solitary wave the integrals

can be evaluated analytically to obtain the specific results of Equation (3.23), with

uL = 0 and uU  = 0. For the present problem with two solitary waves whose

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on the entire spatial interval [−10, 90]. The maximum and minimum node spacings

are approximately 10 and 1/30 times the average node spacing, respectively.

FIGURE 3.12

Collision of two solitary waves leaving a train of disturbances.

FIGURE 3.13

Collision of two solitary waves and details of the train of disturbances.

The invariants of motion at the initial time t  = 0 are calculated in the same man-

ner as for the previous problem, and the final results are given by Cest1 = 0.0,

Cest2 = 129.60 , and Cest

3 = 0.0. The numerical values of the invariants of mo-

tion were evaluated and monitored during the MOL solution procedure, and the value

of |Cnum1 | stayed below 2.81×10−6, Cnum

2 remained constant to five significant digits,

and |Cnum3 | stayed less than 13.9×10−6. Such results provide some assurance that

the numerical computations by the MOL with cfd7po and an adaptive grid are fairly

accurate for this problem.

The third problem for the EW equation with µ = 1 involves the overtaking of one

solitary wave by another, a problem first studied by Abdulloev et al. [1] with the RLW

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equation. The first wave is specified at time t  = 0 by a solitary wave solution given

by Equation (3.2) as 3c1sech2 [k1 (x − x01)] with c1 = ν1 = 3.40, k1 = 0.50, and

x01 = 15, and the second wave is also specified initially by a solitary wave solution as

3c2sech2

[k2 (x − x02)] with c2 = ν2 = 1.70, k2 = 0.50, and x02 = 35. These tworesults are added and used as the initial conditions. The MOL solution is obtained by

using the finite-difference scheme cfd7p6o on an adaptive grid with 301 nodes over

the spatial interval [−10, 130]. Solutions are computed for the time interval [0, 26],and grid adaptation is done at regular time intervals of t grid = 1. The other adaptive

grid parameters are a1 = a2 = 1, α = 0.0001 and β = 0.01. This value of β invokes

large amounts of clipping because the maximum value of |ux | + |uxx | varies from 4

to 7 for the initial waves and their interaction.

A time-distance diagram of the numerical computations is shown in Figure 3.14

for the spatial interval [0, 115], which is the main part of the entire computationalinterval [−10, 130]. The rightward moving wave with the amplitude (3c1 = 10.20)

and speed (ν1 = c1 = 3.40) overtakes the rightward propagating wave with the

smaller amplitude (3c2 = 5.10) and speed (ν2 = c2 = 1.70). After their interaction

they become transmitted waves with almost their original shapes and speeds, but

the trajectories of the large and small amplitude waves are shifted forward (+3) and

backward (−3), respectively. The interaction produces a tiny disturbance whose

location is indicated but its shape is not observable in Figure 3.14. This disturbance

appears stationary from an examination of the numerical results.

FIGURE 3.14

Overtaking of one solitary wave by another leaving a tiny disturbance.

Two spatial distributions at times t  = 0 and 25 are shown more clearly in Fig-

ure 3.15, where the shape of the tiny stationary disturbance is magnified for clarity

(50-fold in amplitude). The distributions of grid nodes at the two different times are

also shown for interest in the lower part of Figure 3.15. The solution by the MOL for

301 nodes provides an excellent solution for this problem, although a good solution

can be obtained with fewer nodes. Only every second node is shown in both parts of 

the figure for clarity. The maximum and minimum node spacings are approximately

4 and 1/3 times the average node spacing, respectively.

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FIGURE 3.15

Overtaking of one solitary wave by another and details of the tiny disturbance.

The invariants of motion are calculated at the initial time t 

=0 in the same manner

as for the previous two problems, and the final results are given by Cest1 = 61.20,

Cest2 = 416.160, and Cest

3 = 2546.89920. The numerical values of the invariants of 

motion were calculated and monitored during the MOL solution procedure, and they

remained constant to five significant digits. These results provide some assurance

that the numerical computations by the MOL with cfd7po and an adaptive grid are

fairly accurate for this problem.

The overtaking of the weaker solitary wave by the stronger solitary wave in this

last problem produces a tiny stationary disturbance that is barely noticeable. Such a

tiny disturbance could be easily overlooked in the results of numerical computations,or it could be mistaken as a numerical effect, and one might then conclude incorrectly

that the interaction is elastic. Historically, this was the first type of inelastic wave

interaction that was investigated, and the study was carried out by Abdulloev et

al. [1] with the RLW equation, despite the numerical resolution dif ficulties with

early numerical methods (e.g., low-order spatial discretization on a uniform grid

and a low-order time integration scheme). Santarelli [26] focused his later studies

with the RLW equation on the collision of two solitary waves, because the inelastic

interaction resulted in a readily observable stationary disturbance in the first problem

of this section and a strong train of disturbances in the second problem, such that

the inelastic behavior could be exhibited clearly with modest computational tools of 

his time. Modern numerical methods based on high-order finite differences with an

adaptive grid and a high-order time integration scheme used in this section clearly

resolve all of these fascinating wave interactions, including the inelastic effects from

a tiny disturbance to a train of strong disturbances.

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3.4.3 Gaussian Pulse Breakup into Solitary Waves

The formation of a train of solitary waves from the breakup, dissolution, or decay

of a single Gaussian shaped pulse specified initially (i.e., at time t 

=0) has fascinated

many researchers working on solutions of the KdV and RLW equations. For the KdV

equation, the initial positive-amplitude Gaussian pulse does not propagate rightward

as a single solitary wave but instead breaks into a train of rightward traveling positive-

amplitude solitons when the value of µ is smaller than some critical value µcrit defined

by Berezin and Karpman [4]. For µ = µcrit, the Gaussian pulse changes shape

slightly andpropagates as a positive-amplitudesolitarywaveandleaves behinda small

oscillatory disturbance. For values of µ larger than µcrit, the Gaussian pulse breaks

into a set of alternating positive- and negative-amplitude waves, forming what might

be a wave packet with one group velocity. Similar wave behavior was discovered for

the RLW equation by Gardner et al. [10]. The initial Gaussian pulse and the formation

of the train of solitary waves have sometimes been referred to as a Maxwellian pulse

or Maxwellian pulses, terminology that is not adopted herein. In this section the EW

equation, ut  + uux − µuxx t  = 0, is solved by the adaptive MOL for the case of an

initial Gaussian pulse, and four different values for µ are chosen to provide a set of 

good illustrations of interesting Gaussian pulse breakups into solitary waves.

For all problems in this section the Gaussian pulse is used to provide the ini-

tial conditions at time t  = 0, having the same symmetric profile u(x,t  = 0) =exp

− (x − x0)2 

centered at the spatial location x0 = 7. This pulse is entirelypositive, has an amplitude of unity, and features a width of 1.66511 at the one-half 

amplitude level. Although the EW equation is solved for four different problems cor-

responding to a wide range of values of µ = 1/100, 1/25, 1/5, and 1, the solutions are

obtained for all four problems for the same spatial domain [−10, 40] and same tem-

poral interval [0, 50] by the MOL using the finite-difference scheme cfd7p6o and an

adaptive grid with 401 nodes. The adaptive-grid parameters are fixed at a1 = a2 = 1,

α = 0.00003, β = 0.01, and t grid = 2 for all four problems, which means they

are not necessarily well tuned for each problem. The Dirichlet boundary conditions

u(xL, t) = uL = 0 and u(xU , t) = uU  = 0 are used in all four problems.The integrals in the invariants of motion given by Equation (3.7) can be integrated

analytically for the initial Gaussian profile specified at time t  = 0. The resulting

constants can be summarized as Cexact1 = √ 

π , Cexact2 = (1+µ)

√ π/2,and Cexact

3 =√ π/3 for later reference. These are evaluated at time t  = 0 and for uL = 0 and

uU  = 0.

Numerical results from the MOL solution of the EW equation for the first problem

with µ = 1/100 are now presented. A time-distance diagram of the computations is

given first in Figure 3.16 for the spatial interval[0, 35

], which is the main part of the

computational interval [−10, 40]. One can see that the initial Gaussian pulse does

not propagate as a single solitary wave. Instead, this pulse breaks up or evolves fairly

rapidly into a set of rightward traveling waves that become more and more separated

in space. The leading wave has the largest amplitude and speed, and the trailing

waves successively decrease in amplitude and speed. These waves, when they are

travelling separately, all appear to behave like solitary waves with thesame theoretical

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FIGURE 3.17

Profiles of solitary waves evolving from an initial Gaussian profile (µ = 1/100).

that the nodes are too widely spaced away from the solitary waves, and the closest

spacing of the nodes throughout the solitary wave train tends to become uniformly

constant at later times and not suf ficiently clustered (see the lowest diagram of thenode spacings). This occurs because the values of  α = 0.00003 and β = 0.01

are somewhat too small for this first problem. For example, the maximum value of 

|ux |+|uxx | ranges from 6 for the Gaussian profile to 80 for the train of narrow solitary

waves at later times, so the low value of β produces too much clipping and prevents

nodes from clustering more appropriately in localized regions of higher gradients and

curvatures associated with individual solitary waves. Increases in the values of α, β

and the number of nodes would provide a better solution resolution and accuracy.

Numerical results from the MOL solution of the EW equation for the second prob-

lem with µ = 1/25 are now given. A time-distance diagram of the numerical com-putations appears in Figure 3.18 for the spatial interval [0, 32], which is the main part

of the computational interval [−10, 40]. The initial Gaussian profile evolves fairly

rapidly into a smaller number of rightward traveling waves than for the first problem,

each successive wave with a smaller amplitude and a slower speed. These waves of 

decreasing amplitude again separate and behave like solitary waves, each with the

same theoretical wave number k = 1/√ 

4µ = 2.5 and consequently the same width.

Although only four waves can be counted in the computational time interval [0, 50],some additional smaller amplitude waves will likely occur at later times, until the

elongating and flattening left side of the Gaussian profile vanishes. On examination

of the numerical results, we find that the constant width of the waves is 0.71 at one-

half amplitude. This width is again smaller than 1.665 for the initial Gaussian profile,

by a factor of 0.42 (as compared to 0.21 for the first problem). The leading wave has

an amplitude 1.31 that is larger than that of the initial Gaussian profile (in contrast to

1.53 for the first problem).

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FIGURE 3.18

Solitary waves evolving from an initial Gaussian profile (µ = 1/25).

Three spatial distributions at times t  = 0, 16, and 32 are depicted in Figure 3.19,

and they once again illustrate the evolution of the Gaussian profile into a sequence of 

waves with decreasing amplitudes and speeds. The elongation and flattening of the

left side of the Gaussian profile can be seen in this figure. From the upper two plots

in Figure 3.19 one can again observe that the peak amplitudes of the train of waves lie

almost on a straight line, and the projected line intersects the x-axis between x

=5 to

6, to the left of the Gaussian profile center (i.e., x0 = 7). When the peak amplitudes

of the trailing waves lie closely on a straight line that includes the leading solitary

wave, these trailing waves are then also solitary waves.

FIGURE 3.19

Profiles of solitary waves evolving from an initial Gaussian profile (µ = 1/25).

From the numerical results displayed in Figure 3.19, the solution by the adaptive

MOL with 401 nodes appears to provide a good solution for the second problem

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From the results shown in Figure 3.21, the MOL solution with 401 nodes provides

a good solution for this third problem in the time interval [0, 40]. This comment is

supported by calculations of the invariants of motion during the MOL computations.

The exact values were reproduced and remained constant to five significant digits.From the upper two diagrams in Figure 3.21, one can see that the grid nodes are

well clustered in regions of high gradients and curvatures, better than for the two

previous problems, because the values of α = 0.00003 and β = 0.01 are now more

appropriate. The maximum value of |ux | + |uxx | ranges from 6 for the Gaussian

profile to 3.5 for the smooth solitary wave and disturbance that evolve later.

 

FIGURE 3.21

Profiles of solitary waves evolving from an initial Gaussian profile (µ = 1/5).

The numerical results from the MOL solution of the EW equation for the fourth and

last problem with µ = 1 are now presented. A time-distance diagram of the numerical

results is given first in Figure 3.22 for the spatial interval [−10, 26], which is the main

part of the computational interval [−10, 40]. The initial Gaussian pulse breaks into a

FIGURE 3.22

Solitary waves evolving from an initial Gaussian profile (µ = 1).

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rightward traveling wave with a positive amplitude and a leftward moving wave with

a negative amplitude, and a small stationary disturbance that remains at x0 = 7. The

initial Gaussian pulse appears to move slowly at first as the leftward and rightward

moving waves form, and then the rightward facing wave moves more quickly. Thetwo waves appear to behave like solitary waves at later times, with the same theoretical

wave number k = 1/√ 

4µ = 1/2 and hence the same width.

Three spatial distributions at times t  = 0, 24, and 48 are displayed in Figure 3.23 to

further illustrate the shape of the leftward and rightward traveling waves. From a close

examination of the numerical data in Figures 3.22 and 3.23, the leftward and rightward

moving waves have different amplitudes of 0.80 and −0.36, respectively, and they

move at speeds of one-third of these values, typical of solitary waves computed from

the EW equation. The widths of the rightward and leftward moving waves at the

one-half amplitude level are the same at about 3.5, which is about twice as large asthat corresponding to the initial Gaussian pulse.

FIGURE 3.23Profiles of solitary waves evolving from an initial Gaussian profile (µ = 1).

The numerical solutions shown in Figures 3.22 and 3.23 for this last problem by

the adaptive MOL with 401 grid nodes are quite good. The node spacings in the

latter figure appear reasonable for the adaptive grid parameters selected for all four

problems. The invariants of motion that were calculated and monitored during the

numerical computations reproduced the theoretical values and remained constant to

five to six significant digits. This enhanced accuracy over the previous three problems

is likely due to the wider and smoother solitary waves and stationary disturbance that

occur in this fourth problem.

Some additional observations can be made from all of the numerical results of the

four problems. When µ ≈ 1/5 the width of solitary waves predicted by the EW

equation is about equal to the width (1.665) of the initial Gaussian pulse. In this

case, the Gaussian pulse evolves into one main solitary wave of about the same width

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and leaves a small stationary disturbance behind. This solitary wave contains the

majority of the mass, momentum, and energy of the initial Gaussian pulse, and the

small remainder is in the small stationary disturbance. When µ < 1/5 the width of 

solitary waves from the EW equation is smaller than that of the initial Gaussian pulse.In this case a train of narrow solitary waves must evolve from the initial Gaussian

pulse to account for all of its initial mass, momentum, and energy. For smaller values

of µ and correspondingly larger mismatches in the widths of the solitary waves and

Gaussian pulse, the train of narrower solitary waves is more numerous and more

closely spaced. When µ > 1/5 the width of solitary waves from the EW equation is

larger than that of the initial Gaussian profile. In this case the single, wide solitary

wave contains more mass, momentum, and energy than the initial Gaussian pulse,

and this seemingly results in a negative-amplitude pulse that becomes the provider

of the extra mass and energy, although some may remain in the small disturbanceremaining between the waves.

From the numerical results for the four problems, we deduce that the critical value

of µ is roughly 1/5, and this criterion determines whether the breakup of the Gaussian

pulse leads to a train of solitary waves (µ < 1/5), only one solitary wave with a small

stationary disturbance (µ ≈ 1/5), or a leftward moving solitary wave and a rightward

moving solitary wave separated by a small disturbance (µ > 1/5). However, this

critical value of µ can be determined approximately by using the conservation laws.

Let us equate the mass√ 

π and momentum (1+

µ)√ 

π/2 in the Gaussian pulse to the

corresponding mass 12c√ µ and momentum (144/5)c2√ µ of a solitary wave from

the EW equation. These equivalences result in two nonlinear algebraic equations

for the two unknowns µ and c, and the solution by means of an analytic or iterative

method is given by µestcrit = 0.1804 and cest

crit = 0.3478 to four significant digits. If 

the mass and energy are used instead, then the critical values are very similar and

given by µestcrit

= (π/30)√ 

3 = 0.1814 and cestcrit

= (25/1728)1/4 = 0.3468. These

critical values are estimates only, because the solitary wave that evolves from the

initial Gaussian pulse cannot contain exactly all of the mass, momentum, and energy

of the initial Gaussian pulse.

3.4.4 Formation of an Undular Bore

A long wavewith a gradually and monotonically sloped front canpropagate in deep

water without significant change in shape, when the nonlinear effects ofsteepeningare

balanced by dissipation and dispersion. However, as a long wave travels into shallow

water, the smoothly varying front can steepen further, and this type of a steepening

wave is called a bore. Ocean tides can produce large bores (e.g., amplitude of 3 m)

that propagate upstream in river channels and attract bore watchers. When the surface

elevation of the water behind a long bore is less than 0.28 times the water depth in

front, the steepening front of a bore that is initially smoothly varying will develop

surface ripples that grow into a train of large oscillations or undulations, and this

type of a wave is called an undular bore. See Peregrine [23] for more details. In this

section the EW equation, ut + uux − µuxx t  = 0, with µ = 1/6 is solved by the MOL

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with an adaptive grid for a problem involving the development of an undular bore.

The value of 1/6 is taken for µ so that the EW equation becomes applicable to the

case of water waves and bores.

The initial conditions for the numerical computations are specified at time t  = 0by u(x,t  = 0) = (1/2)u0 [1 − tanh (X)], with the nondimensional distance X =(x − x0)/d . The spatial distribution of this initial bore is monotonically decreasing

from an asymptotically constant amplitude u = u0 as x → −∞ to the one-half 

amplitude level u = u0/2 at x = x0, where the surface slope ux = −(u0/2)/d , and

then decreasing to the asymptotically constant amplitude u = 0 as x → +∞. The

parameters for this initial bore shape are given by the bore amplitude u0 = 1/10, the

initial location x0 = 0 of the center of the bore (at half amplitude) and slope control

parameter d 

=5. These are the same bore parameters that were used in a previous

solution by Gardner et al. [10].

The problem involving the formation of an undular bore is solved in this study by

the MOL using the finite-difference scheme cfd7p6o and an adaptive grid with 401

nodes, and for the spatial and temporal intervals [−20, 55] and [0, 800], respectively.

The adaptive grid parameters are a1 = a2 = 1, α = 0.001, β = 10 (no clipping),

and t grid = 1. The Dirichlet boundary conditions uL = u0 and uU  = 0 are applied

at the lower and upper grid boundaries.

The integrals in the invariants of motion given by Equation (3.7) can be inte-

grated analytically or numerically for the initial smooth bore profile specified att  = 0. The resulting invariants of motion, for the case of uL = u0 and uU  = 0,

are summarized for later reference as Cexact1 = 2.000084, Cexact

2 = 0.1751279, and

Cexact3 = 0.01625252.

Numerical results from the MOL solution of the EW equation with µ = 1/6 are

given first in the form of a time-distance diagram in Figure 3.24 for the reduced spa-

tial interval [−20, 46], which is the main part of the entire computational domain

[−20, 55]. The front of the initially smooth bore begins to steepen as it propagates to

the right, and this front eventually breaks into an ever increasing number of undula-

tions, one after the other, forming what is called an undular bore. This bore continuesto advance in space and time as a train of oscillatory waves. A close observation of 

the formation of each undulation will reveal that its peak amplitude increases from

an initial value of 0.10 for the initial bore to a somewhat larger amplitude at larger

distances and times. This train of undulatory waves carries the mass, momentum,

and energy of the initial bore forward in space and time.

A clearer view of the cross-section of the undular bore is presented in Figure 3.25,

where three spatial distributions at the times t  = 0, 300, and 600 are depicted. A

large number of nodes is required to obtain an accurate solution at later times because

of the increasing number of undulations. For the computational time interval of 

[0, 800], the use of 401 grid nodes is more than suf  ficient, as can be seen by the

node distribution in the undulations at the later time of t  = 600. The grid nodes

are well clustered in regions of large gradients and curvatures. Note that only every

second node is shown for clarity. The invariants of motion were calculated during the

numerical computations to help indicate if the solution is computed accurately. The

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FIGURE 3.24Formation of an undular bore.

exact values were reproduced and remained constant to four significant digits. This

provides more assurance that the solution is computed accurately.

FIGURE 3.25

Cross-sections through an undular bore.

The increase in peak amplitude with time of the leading wave of the undular bore

and the corresponding slightly concave trajectory are both shown in Figure 3.26. The

peak amplitude of this leading wave can be clearly seen to increase rapidly at first

from the initial value of 0.10 of the original bore, and then it rises more slowly to

what appears as an asymptotic limit that is at least 0.182. The trajectory of the peak 

amplitude of the leading wave accelerates, more quickly at smaller times and then

asymptotically to a final speed. The slope of the trajectory at later times, which is the

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FIGURE 3.26

Amplitude growth and trajectory of the leading wave of an undular bore.

asymptotic speed of the leading wave, is about 0.061, which is about one-third of the

peak amplitude.

A close inspection of the numerical data for the leading undulation shows that

its shape and speed are consistent with those of a solitary wave. The following

undulations have similar shapes and behaviors, but they are closely and uniformly

spaced. These undulations do not separate as they propagate, as we observed for the

earlier case of the breakup of an initial Gaussian pulse into a train of solitary waves.

The numericalcomputationsby the MOL for the undular borewere rather computer

intensive on the spatial and temporal domains of [−20, 55] and [0, 800], as compared

to all of the previous problems in this study. The CPU time was 142 min on a

Pentium III computer with a 500-MHz processor and LINUX operating system. For

the adaptive grid with 401 nodes, the total number of time steps was 23430, the

number of function evaluations was 35514, and the number of Jacobian evaluations

was 12585.

3.5 Concluding Remarks

An advanced numerical MOL for solving the EW equation on uniform and adaptive

grids with different finite-difference schemes (cfd7p6o and cfd5p4o) and various

numbers of nodes was explained in detail in this study, primarily to help newcomers to

the MOL learn these techniques more easily and quickly. Many interesting graphical

solutions were computed by means of the MOL for the first time and presented to

illustrate the fascinating behavior of a single solitary wave, the inelastic interaction

of two solitary waves, the breakup of a Gaussian pulse into a sequence of solitary

waves, and the formation of an undular bore. An equal emphasis was placed on the

description of the numerical MOL subprocedures and on the illustration of interesting

numerical results by means of informative diagrams.

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The numericalMOL solutions presented in this study for the EW equation are more

accurate and computed more ef ficiently than those in previous work on the EW and

RLW equations for the same problems. These advantages stem primarily from the

use of an adaptive grid with a high-order finite-difference scheme (cfd7p6o) and ahigh-order time integration(DASSL) of the DAEs, and also from enhanced techniques

of interpolation of numerical data during grid adaptations.

Three objectives of this study were to implement improvements in some subproce-

dures of the numerical MOL. The first objective was to provide an improved technique

of interpolating numerical data than the previous usage of cubic and quintic splines.

The mapping of data by either spline interpolation from a previous to a newly adapted

grid normally adversely reduces the accuracy of the numerical solution, although the

solution may be smoothed to some advantage by the spline interpolation. This low-

order spline interpolation is counterproductive to improving the solution accuracy

by the implementation of high-order finite-difference schemes associated with large

stencils in the MOL. In this study we interpolated numerical data by means of the

quintic polynomial given by Equation (3.21), which we believe is an improvement

in both accuracy and computer ef ficiency as compared to cubic and quintic splines.

However, the underlying polynomial equation (e.g., Lagrange) associated with the

finite-difference scheme should have been used for the interpolations. For example,

the finite-difference scheme labeled cfd7po has a seven-point stencil with an asso-

ciated sixth degree polynomial equation through seven data pairs (u vs. x). Thisunderlying polynomial can be implemented readily for interpolation, and the accu-

racy of such an interpolation is then inherently consistent with the finite-difference

scheme in the MOL procedure.

The second objective was to provide an improved technique of integrating numeri-

cal data than the previous approach of using Simpson’s integration method. Accurate

integrations are required, for example, to calculate the three invariants of motion, and

Simpson’s method is normally inadequate in terms of accuracy. In this study, the

integrals in the invariants of motion were evaluated by using the quintic polynomialgiven by Equation (3.21), which yields suf ficiently accurate invariants of motion.

However, the underlying polynomial equation associated with the finite-difference

scheme should have also been used for these integrations. This underlying polyno-

mial can be used easily for the integrations, and the accuracy of the quadrature is then

inherently consistent with the finite-difference scheme in the MOL procedure.

The third objective of this study was to improve the method of selection of values

of some parameters for adaptive gridding. The use of a static adaptive grid technique

in the MOL was found to be both cumbersome and time consuming for the user of 

the computer code. The selection of optimal values for the parameters a1, a2, α, β,

γ , and t grid for a particular problem was not only tedious, but it also depended to a

large extent on subjective judgement. The description presented in this study on how

to select appropriate values for α and β helps to simplify the selection process by

illustrating their interdependence and also their relationship to gmax and gmin, which

are multiples of the average grid node spacing.

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The selection process for values of the parameters a1, a2, α, β, γ , and t grid should,

regardless of our helpful improvements, be fully automated and simply made part of 

an adaptive grid package. Such an automated process should be linked, hopefully,

to the equidistribution of truncation errors throughout the grid. Adaptive gridding iscurrently implemented after a given time interval t grid, and then the time integration

in the solver DASSL is restarted with the order q = 1, progressing to higher orders

up to q = 5 at successive time steps. On one hand, if the value of t grid is set

too small, the numerical results are less accurate due to the degraded order of the

time integration. On the other hand, if t grid is set too large, then the waves outrun

their regions of clustered nodes and the numerical results become less accurate. This

problem can be overcome by implementing a high-order solver that advances the

solution in time of the stiff and implicit DAEs by means of a one-step method.

Acknowledgments

The financial support from the Natural Sciences and Engineering Research Coun-

cil of Canada for Professors J.J. Gottlieb (Grant No. OGP0004539) and J.S. Hansen

(GrantNo. OGP0003663) is gratefully acknowledged. We express our deepest appre-

ciation to Professor W.E. Schiesser of Lehigh University at Bethlehem, Pennsylvania,

and Professor J.L. Bona of the University of Texas at Austin, Texas, for helpful sug-

gestions.

References

[1] Kh.O. Abdulloev, H. Bogolubsky, and V.G. Markhankov, One more example

of inelastic soliton interaction, Phys. Lett. A, 56, 427–428, 1976.

[2] U.M. Ascher, and L.R. Petzold, Computer Methods for Ordinary Differential

 Equations and Differential-Algebraic Equations, SIAM, Philadelphia, 1988.

[3] T.B. Benjamin, J.L. Bona, and J.L. Mahoney, Model equations for long waves

in nonlinear dispersive media, Phil. Trans. Roy. Soc. Lond. A, 272, 47–78, 1995.

[4] Yu. Berezin and V.I. Karpman, Nonlinear evolution of disturbances in plasmas

and other dispersive media, Soviet Physics JETP, 24, 1049–1056, 1967.

[5] J.L. Bona, W.G. Pritchard, and L.R. Scott, Numerical schemes for a model for

nonlinear dispersive waves, J. Comput. Phys., 60, 167–186, 1985.

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[6] K.E. Brenan, S.L. Campbell, and L.R. Petzold, The Numerical Solution of 

 Initial Value Problems in Differential-Algebraic Equations, Elsevier Science

Publishing Co., 1989.

[7] R.P. Brent, Algorithms for Minimization without Derivatives, Prentice-Hall,

Englewood Cliffs, NJ, 1973.

[8] B. Fornberg, Calculation of weights in finite difference formulas, SIAM Rev.,

40(3), 685–691, 1998.

[9] G.E. Forsythe, M.A. Malcolm, and C.B. Moler, Computer Methods for Math-

ematical Computations, Prentice-Hall, Englewood Cliffs, NJ, 1977.

[10] L.R.T. Gardner, G.A. Gardner, A. Ayoub, and N.K. Amein, Simulations of the

EW undular bore, Commun. Numer. Meth. Engng., 13, 583–592, 1997.

[11] L.R.T. Gardner, G.A. Gardner, and A. Dogan, A least-squares finite element

scheme for the RLW equation, Commun. Numer. Meth. Engng., 12, 795–804,

1996.

[12] L.R.T. Gardner and G.A. Gardner, Solitary waves of the regularized long-wave

equation, J. Comput. Phys., 91, 441–459, 1990.

[13] C.W. Gear, The simultaneous numerical solution of differential-algebraic equa-

tions, IEEE Trans. Circuit Theory, CT–18:89–95, 1971.

[14] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and 

 Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991.

[15] S. Hamdi, J.J. Gottlieb, and J.S. Hansen, Computation of blow-up solutions of 

the generalized Korteweg-de Vries equation using adaptive method of lines, in

J. Militzer, ed., 7th Annual Conference of the CFD Society of Canada, 921–926,

June 1999.

[16] S. Hamdi, J.J. Gottlieb, and J.S. Hansen, Numerical solution of the equal widthwave equation using the method of lines, in 8th Annual Conference of the CFD

Society of Canada, 129–134. Centre de Recherche en Calcul Appliqué, 2000.

[17] A.C.Hindmarsh,ODEPACK: A systematizedcollectionofODE solvers, inR.S.

Steplemman, ed., Scientific Computing, 55–64. North-Holland, Amsterdam,

1983.

[18] P.C. Jain, R. Shankar, and T.V. Singh, Numerical solutions of regularized long

wave equation, Commun. Numer. Meth. Engng., 9, 579–586, 1993.

[19] D.J. Korteweg and G. de Vries, On the change of form of long waves advancing

in a rectangular canal and a new type of long stationary wave, Phil. Mag., 39,

422–443, 1895.

[20] J.C. Lewis and J.A. Tjon, Resonant production of solitons in the RLW equation,

Phys. Lett. A, 73, 275–279, 1979.

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Chapter 4

 Adaptive Method of Lines for Magnetohydrodynamic PDE Models

P. A. Zegeling and R. Keppens

4.1 Introduction

An adaptive grid technique for use in the solution of multi-dimensional time-

dependent PDEs is applied to several magnetohydrodynamic model problems. Thetechnique employs the method of lines and can be viewed both in a continuous and

semi-discrete setting. By using an equidistribution principle, it has the ability to track 

individual features of the physical solutions in the developing plasma flows. More-

over, it can be shown that the underlying grid varies smoothly in time and space.

The results of several numerical experiments are presented which cover many aspects

typifying nonlinear magneto-fluid dynamics.

Many interesting phenomena in plasma fluid dynamics can be described within the

framework of magneto-hydrodynamics (MHD). Numerical studies in plasma flows

frequently involve simulations with highly varying spatial and temporal scales. As aconsequence, numerical methods on uniform grids may be inefficient to use, since a

very large number of grid points is needed to resolve the spatial structures, such as

shocks, contact discontinuities, shear layers, or current sheets. For the efficient study

of these phenomena, we require adaptive grid methods which automatically track and

spatially resolve one or more of these structures.

Over the years a large number of adaptive grid methods have been proposed for

time-dependent PDE models. Two main strategies of adaptive grid methods can

be distinguished, namely, static-regridding methods and moving-grid or dynamic-

regridding methods. In static-regridding methods (denoted by h-refinement) the lo-

cation of nodes is fixed. A method of this type adapts the grid by adding nodes

where they are necessary and removing them when they are no longer needed. The

refinement or de-refinement is controlled by error estimates or error monitor values

(which have no resemblance with the true numerical error). Recent examples of these

methods are described in [16, 4, 20, 7]. In dynamic-regridding methods (denoted by

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r-refinement) nodes are moving continuously in the space-time domain, like in classi-

cal Lagrangian methods, and the discretization of the PDE is coupled with the motion

of the grid. Examples can be found in [3, 18, 19, 5, 8].

In this chapter we follow the second approach. The adaptive grid method is basedon a semi-discretization of a fourth-order PDE for the grid variable and is being

coupled to the original MHD model re-written in a new coordinate system. We use

the so-called method-of-lines technique (MOL) [11]: first we discretize the PDEs in

the space direction using a finite-difference approximation, so as to convert the PDE

problem into a system of stiff, ordinary differential equations (ODEs) with time as

independent variable. The discretization in time of this stiff ODE system then yields

the required fully discretized scheme.

The layout of the chapter is as follows. In the next section we present the full

set of MHD equations and their physical meaning. In Section 4.3 we describe therestriction to the one-dimensional situation and the adaptive grid method. The mov-

ing grid is defined as the solution of an adaptive grid PDE. Numerical experiments

are shown for three different cases: an MHD-shocktube model, a problem describing

Shear-Alfvén wave propagation, and an oscillating plasma sheet in vacuum surround-

ings. Section 4.4 discusses the essential elements for generalizing the MOL approach

to multi-dimensional MHD simulations. We evaluate different means for 2D grid

adaptation on kinematic magnetic field models, with particular attention paid to the

solenoidal condition on the magnetic field vector. Section 4.5 lists our conclusions

and presents an outlook to future work.

4.2 The Equations of Magnetohydrodynamics

The MHD equations govern the dynamics of a charge-neutral “plasma.” Just like

the conservative Euler equations provide a continuum description for a compressible

gas, the MHD equations express the basic physical conservation laws to which a

plasma mustobey. Because plasma dynamics is influenced bymagnetic fields through

the Lorentz-force, the needed additions in going from hydrodynamic to magneto-

hydrodynamic behavior is a vector equation for the magnetic field evolution and

extra terms in the Euler system that quantify the magnetic force and energy density.

Using the conservative variables density ρ, momentum density m ≡ ρv (with

velocity v), magnetic field B, and total energy density e, the ideal MHD equations

can be written as follows (cfr. [2, 13, 15]):

Conservation of mass:

∂ρ

∂t + ∇ · (ρv) = 0 . (4.1)

Conservation of momentum:

∂(ρv)

∂t + ∇ · (ρvv − BB) + ∇ pt ot  = 0 . (4.2)

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Conservation of energy:

∂e

∂t + ∇ · (ev + vpt ot  − BB · v) = 0 + m(∇ × B)2 . (4.3)

Magnetic field induction equation:

∂B

∂t + ∇ · (vB − Bv) = 0 [+ mB] . (4.4)

In (4.2) and (4.3) the total pressure ptot consists of both a thermal and a magnetic

contribution as given by

ptot = p +B2

2 , where p = (γ  − 1)

e − ρ

v2

2 −B2

2

(4.5)

is the thermal pressure. This set of equations must be solved in conjunction with an

important condition on the magnetic field B, namely the non-existence of magnetic

“charge” or monopoles. Mathematically, it is easily demonstrated that this property

can be imposed as an initial condition alone, since

∇ · B|t =0 = 0 ⇒ ∇ · B|t ≥0 = 0 . (4.6)

In multi-dimensional numerical MHD, the combined spatio-temporal discretizationmay not always ensure this conservation of the solenoidal character of the vector

magnetic field. When dealing with a two-dimensional model problem for B evolution

in Section 4.4.2.3.3, we pay particular attention to this matter.

The terms between brackets in Equations (4.3) and (4.4) extend the ideal MHD

model with the effects of Ohmic heating due to the presence of currents. With the

resistivity m = 0, we then solve the resistive MHD equations. Likewise, extra

non-conservative source terms may be added to the momentum and energy equation

for describing viscous effects. In the numerical experiments, we resort to artificial

diffusive terms which can be thought of as approximations representing these actualphysical phenomena.

4.3 Adaptive Grid Simulations for 1D MHD

4.3.1 The MHD Equations in 1D

If we restrict the MHD model (4.1)–(4.6) to 1.5D, i.e., variations in one spatial

x-dimension but possibly non-vanishing y-components for the vector quantities with

∂/∂y = 0, we obtain a 5-component PDE system which is formally written as

∂

∂t + ∂F()

∂x= 0, x ∈ [xL, xR] , t > 0 . (4.7)

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Here, = (ρ,m1, m2, B2, e)T  is the vector of conserved variables (m1, m2 are now

the x- and y-components of the momentum vector and B2 denotes the y-component

of the magnetic induction), with the flux-vector F = (F 1, . . . , F  5)T  given by

F 1 = m1 ,

F 2 = m21

ρ− B2

1 + (γ  − 1)e − (γ  − 1)m2

1 + m22

2ρ+ (2 − γ )

B21 + B2

2

2,

F 3 = m1m2

ρ− B1B2 ,

F 4 = B2m1

ρ− B1

m2

ρ,

F 5 = m1ρ

γ e − (γ  − 1) m

2

1 + m

2

22ρ

+ (2 − γ ) B

2

1 + B

2

22

− B1

B1

m1

ρ+ B2

m2

ρ

.

The constant γ  is the ratio of specific heats and B1 is the constant first component

of the magnetic induction vector. Indeed, in 1D model problems, the vanishing

divergence of the magnetic field is thereby trivially satisfied. The remaining set of 5

PDEs given by (4.7) constitutes the physical model used for the 1D MHD simulationsfound below. We first indicate how this system is further manipulated and discretized

to solve simultaneously for the adaptive grid with its corresponding solution.

4.3.2 The Adaptive Grid Method in One Space Dimension

4.3.2.1 Transformation of Variables

It is common practice in adaptive grid generation to submit the PDE model to a

coordinate transformation. Ideally, the mapping should be chosen such that in the

new coordinate variables, the discretization error in the numerical solution is much

smaller than in the original variables. In the new variables the PDEs are then simply

uniformly partitioned. In general, applying a transformation

ξ  = ξ(x,t) ∈ [0, 1], θ  = t , (4.8)

to the system (4.7) gives after some elementary calculations

xξ θ 

−ξ xθ 

+(F ())ξ 

=0 . (4.9)

Different choices for the transformation are possible. The coordinate transformation

used in this chapter is implicitly defined as the solution of a special partial differential

equation (see Section 4.3.2.2). Even without knowing this mapping, we can already

semi-discretize (4.9) by noting that in the ξ -variable a uniform grid (ξ i = i/N,i =0, . . . , N  ) is imposed. Using central finite differences, the PDEs (4.9) become a

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system of ODEs as follows:

(xi+1 − xi−1)di

dθ − (i+1 − i−1)

dxi

dθ + F (i+1) − F (i−1) = 0 ∀i .

(4.10)

Note that we have multiplied (4.10) by the factor 2ξ , which has a constant value by

definition.

4.3.2.2 The Adaptive Grid PDE

We implicitly define the transformation ξ(x,t), and thereby the grid distribution,

as the solution of the following time-dependent “adaptive grid PDE”

S 

xξ + τ xξ θ 

W 

ξ  = 0 . (4.11)

The parameter τ > 0 in (4.11) is a temporal smoothing parameter, the operator S 

incorporates a spatial smoothing in a manner detailed below, while

W =

1 +5

j =1

αj 

(j )x

2

is a weight function that depends on the derivatives of the different components(j ).

The parameters αj  are termed “adaptivity parameters.” Their values can be chosen

to emphasize, if necessary, particular variables in the PDE model (such as the density

or a magnetic field component for MHD problems). In full, the smoothing operator

S  in (4.11) is defined by

S =  I − σ (σ  + 1)(ξ)2 ∂2

∂ξ 2, (4.12)

where σ > 0 is a spatial smoothing parameter and I  the identity operator. This

specific choice of transformation has several desirable properties, which are brieflydiscussed in Section 4.3.2.3.

Since the adaptive grid PDE is fourth order in space, it is clear that we need four

boundary conditions and one initial condition. An obvious choice is to take two

Dirichlet and two Neumann conditions:

x|ξ =0 = xL, x|ξ =1 = xR , xξ |ξ =0 = xξ |ξ =1 = 0 . (4.13)

At initial time θ  = 0, the grid is uniformly distributed and is thus given by x|θ =0 =xL

+(xR

−xL)ξ .

4.3.2.3 Properties of the Adaptive Grid

It can be shown that the determinant of the Jacobian of the transformation implied

by (4.11) and (4.12) satisfies the mesh-consistency condition

J  = xξ  > 0 ∀ θ  ∈ [0, T ], ∀ξ  ∈ [0, 1] , (4.14)

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which in discretized form reads (since ξ  is a constant)

xi (θ ) := xi (θ ) − xi−1(θ ) > 0 ∀ θ  ∈ [0, T ] . (4.15)

In other words, relation (4.15) states that the grid points can never cross one another

(see Chapter 4 in [18] and [5] for more details and proofs of these results). Another

important property of the transformation satisfying (4.11) and (4.12) is the following:xξ ξ 

xξ 

≤ 1√ σ (σ  + 1)ξ 

, (4.16)

which may be translated in discrete terms as

1

1 + 1σ 

≤ xi+1(θ )

xi (θ )≤ 1 + 1

σ ∀ θ  ≥ 0, ∀i . (4.17)

This property expresses “local quasi-uniformity” and means that the variation in

successive grid cells can be controlled by the parameter σ  at every point in time.

A reasonable choice for the temporal smoothing parameter is 0 < τ  ≤ 10−3 ×{timescale in PDE model}, while the spatial smoothing parameter is typically σ  =O(1). The adaptivity parameters are normally taken αj  = O(1) (see also [18]), but

may need re-scaling depending on the x-range and the magnitude of the individual(j ). Note that if we switch off all smoothing in (4.11), we obtain the well-known

“equidistribution principle” which has both a continuous and discrete variant given

by the formulae

τ  = σ  = 0 ⇒ xξ  W 

ξ 

= 0 ∀ θ  ∈ [0, T ] ⇔ ξ(x,t) = x

xLW  dx xR

xLW  dx

,

(4.18)

or in discretized form (using the midpoint rule for integration)

xi ·W i−1/2 = constant ∀ θ  ∈ [0, T ] . (4.19)

4.3.2.4 Semi-Discretization of the Adaptive Grid PDE

Theadaptivegrid PDE(4.11) is semi-discretized usingcentral-differences to obtain

xi

+1

+τ 

dxi+1

dθ W i+1/2

− xi

+τ 

dxi

dθ W i−1/2

=0

∀i , (4.20)

where

xi = xi − σ (σ  + 1) (xi+1 − 2xi + xi−1) , (4.21)

which is a discretization of S (xξ ) about the gridpoint ξ i = iN 

.

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The discrete version of the weight function becomes

W i−1/2 = 1 +

5j =1

αj (

(j )

x,i−1/2)2

, with

(j )

x,i−1/2 =

(j )

i

−

(j )

i

−1

xi . (4.22)

The grid equations (4.20) and (4.21) are related to the adaptive grid described in

[3]. The differences between (4.20) and (4.21) and [3] mainly consist of the use of 

cell-lengths instead of point concentrations and not applying the operator S  to thedxdθ 

-terms. In compact notation the adaptive grid ODE system (4.20) reads

τ  B (X,, σ, α)dX

dθ  =H(X,, σ, α) , (4.23)

where

α = (α1, α2, . . . , α5)T  ,

and and X contain the discretized MHD components and the grid points, re-

spectively. After coupling this system to the semi-discretized PDE system (4.10),

a large, stiff, banded, nonlinear ODE system is obtained. System (4.23) has band-

width 12. This can be derived easily by working out (4.20) in terms of the xi’s

and realizing that the unknown vector of the complete ODE system is written as

( . . . ,(1)i ,(2)i , . . . ,(5)i , xi ,(1)i+1, . . . )T  . For the time-integration of this system,the ODE-package DASSL [10] with the (implicit) BDF-methods up to order 5 will be

used. DASSL uses a direct solver for the linear systems and exploits the banded form

of the equations in the Jacobian formation and numerical linear algebra computations.

Numerical differencing for Jacobians in the Newton-process is being used. The time-

stepping error tolerance is denoted by tol and will be specified at the experiments.

4.3.3 Numerical Results

In what follows, we apply the adaptive MOL approach to three 1.5D MHD modelproblems which are chosen to cover significantly diverse challenges typically en-

countered in numerical MHD simulations. We solve a standard Riemann problem

to address the performance of the MOL technique as a shock-capturing and shock-

tracing method, we simulate linear shear Alfvén waves which are non-compressive

perturbations with a specific polarization, and we model a plasma-“vacuum” con-

figuration which poses numerical difficulties to keep density and pressure positive

throughout the domain. We explicitly compare the obtained adaptive grid solu-

tions with high resolution reference solutions on static, uniform grids. These ref-

erence solutions are all calculated with the Versatile Advection Code [14] (VAC,

see http:// www.phys.uu.nl/∼toth), and if not stated otherwise, use 1000 grid

points and the (approximate)Riemann-solver basedtotalvariation diminishing(TVD)

scheme with “minmod” limiting. This shock-capturing, one-step TVD scheme is ac-

tually one out of six high resolution spatial discretization schemes available in VAC,

and has demonstrated to be the most accurate and efficient discretization method on

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a large variety of HD and MHD problems [13]. Specifically, the effects of dispersion

and diffusion will essentially be minimal in the reference solutions, and this should

be kept in mind when comparing them with the adaptive simulation results. For the

latter, all experiments used values for the smoothing parameters τ  = 10−5

, σ  = 2,and tol = 10−6. The adaptivity constants will be specified and motivated per model.

4.3.3.1 MHD-Shocktube Model

This test problem by [2] and also used in [13] has evolved into a benchmark for

MHD codes. The initial Riemann problem separates a high density and high thermal

pressure left state from a low density and low pressure right state with the magnetic

field lines reflected over the normal to the discontinuity line x = 0.5 in the x − y

plane. The sudden expansion of the left state produces a reversedly propagating fast

rarefaction fan and a slow compound wave, a rightwardly advected contact disconti-

nuity, and a right-moving slow shock and fast rarefaction fan. The compound wave

is a combination of a slow shock with a slow rarefaction attached to it.

Specifically, theproblem is setup in thespace-interval x ∈ [0, 1], while we simulate

for times t  ∈ [0, 0.1]. In the adaptive approach, we use 250 grid points, and added

artificial diffusion terms  1J  

∂∂ξ 

[ 1J  

∂(j )

∂ξ ] to all but the mass-conservation law with

diffusion coefficients = 0.0001. Since all developing dynamic features have an

associated density variation, we set the adaptivity parameter α1 = 1, while all other

parameters αi = 0, (i = 2, . . . , 5). Furthermore,

γ  = 2, B1 ≡ 0.75

ρ|t =0 =

1 for x ∈ [0, 0.5]0.125 for x ∈ [0.5, 1]

m1|t =0 = m2|t =0 = 0

B2|t =0 = 1 for x ∈ [0, 0.5]

−1 for x

∈ [0.5, 1

]e|t =0 =

1.78125 for x ∈ [0, 0.5]0.88125 for x ∈ [0.5, 1]

Homogeneous Neumann boundary conditions are used for all components. In

Figure 4.1, we compare the density profile at t  = 0.1 from three simulations: a VAC

solution on a 250-point static grid, the adaptive solution with the same amount of 

grid points, and the true reference VAC solution exploiting 1000 points (both VAC

solutions used a Courant number of 0.8). Clearly, the MOL technique is superior to

the VAC solution that uses the same amount of grid points, and the accuracy of the

adaptive method is identical to the high resolution reference solution. We thus save

a factor of 4 in grid resolution as compared to a uniform grid. In Figure 4.2, we plot

at left the v1 := m1/ρ velocity profile at the same time for both the adaptive and

the reference solution, while the grid history for t  ∈ [0, 0.1] is shown at right. Note

that the adaptive solution is fairly dispersive for this particular variable. The grid

history demonstrates how the initial discontinuity causes an immediate clustering of 

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grid points in the region of interest and that the emerging shock features are nicely

traced individually.

FIGURE 4.1

Density at t  = 0.1 for the magnetic shocktube model. We compare two static

grid reference solutions, one with 250 grid points (dots) and one for 1000 grid

points (dashed), with an adaptive MOL solution exploiting 250 points (solid).

FIGURE 4.2Left panel: v1 component of the velocity t  = 0.1 for both the reference (dashed)

and the MOL solution (solid). Right panel: grid history (tracing x-positions of 

grid points as a function of time t ) for the magnetic shocktube model.

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4.3.3.2 Shear-Alfvén Waves

This test problem was described by [12] and also used in [13] for their evaluation

of different discretization schemes. A homogeneous, uniformly magnetized plasma

state is perturbed with a localized velocity pulse transverse (v2 := m2/ρ = 0) to thehorizontal (x-direction) magnetic field. This evolves into two oppositely traveling

Alfvén waves that only have associated v2 := m2/ρ and B2 perturbations. The

complete problem setup is as follows.

We take x ∈ [0, 3] and time-interval t  ∈ [0, 0.8], together with artificial diffusion

coefficients (except for the mass-conservation law) = 0.0001. Because we only

expect transverse vector components, we set the adaptivity parameters α3 = α4 =1e + 8 with all other α1 = α2 = α5 = 0. The high values for α3 and α4 are a

consequence of a scaling effect in the weight function. Since (B2

x

)2

=O(10

−8)

occurs in W , it is natural to choose the adaptivity parameter(s) O(108) to balance

the different terms. The number of grid points for this model is taken equal to 250.

Physical parameters and initial conditions for this model are:

γ  = 1.4, B1 ≡ 1

ρ|t =0 = 1

m1

|t =

0

=0

m2|t =0 =

10−3 for x ∈ [1, 2]0 elsewhere

B2|t =0 = 0

e|t =0 =

0.5000005025 for x ∈ [1, 2]0.5000000025 elsewhere

Homogeneous Neumann boundary conditions hold for all components.

Figure 4.3 shows the B2 component of the magnetic induction at t  = 0.8 from boththe MOL and the reference solution. In the right panel, the grid history is shown.

The solution again compares favorably to the high resolution static grid simulation,

only slightly worsened by dispersion. The grid history shows how the original single

pulse separates into two oppositely traveling signals. In Figure 4.4, we compare the

errors present for both the reference solution and the adaptive one: ideally the density

should remain constant. Noting the large difference in scales, the MOL approach

succeeds better in minimizing the density variations. In fact, we used a Courant

number of 0.4 for the reference solution in order to suppress these errors somewhat.

For the reference result, they are due to the small thermal pressure (p = 10−9) which

creates roundoff problems within the Riemann solver used (see also [13]). Indeed,

when switching to the non-Riemann solver based TVD Lax-Friedrichs discretization

in VAC, these errors essentially disappear. Although the MOL solution seems better

 judged from the controlled density variations, it fails to maintain the positivity of the

thermal pressure for this example.

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FIGURE 4.3

Left panel: y-component of magnetic induction for the Shear-Alfvén problem

at t  = 0.8, again from a 1000-point reference solution (dashed) with a 250 MOL

solution. Right panel: grid history for the adaptive simulation.

FIGURE 4.4

Comparison of the errors in the density profile for the reference MOL approach

(left) and the TVD result (right). Note the different scales on the ρ-axes.

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4.3.3.3 Oscillating Plasma Sheet

This test model was introduced recently in [15] as a typical case where an implicit

time integration strategy is more efficient than explicit methods. A sheet of high den-

sity and pressure is surrounded by a magnetized “vacuum.” The vacuum is modeledas a low density, low pressure plasma so that the plasma-vacuum interface is prone

to introduce non-physical negative density and/or pressure fluctuations. We set up

an initial total pressure imbalance across the sheet by prescribing a uniform, sheet-

aligned magnetic field of different magnitude in the left and right vacuum region.

With ideally conducting wall boundary conditions at some distance away from the

sheet boundaries, this results in a magnetically controlled oscillation of the sheet as a

whole due to alternate compressions and rarefactions of the vacuum magnetic fields

on either side.

Specifically, for x ∈ [0, 1], time t  ∈ [0, 2], we now use artificial diffusion co-

efficients = 0.001 for momentum, energy, and magnetic field, while it was even

necessary for stability reasons to introduce an artificial diffusion term in the mass-

conservation law with = 10−5. We took as adaptivity parameters αi = 1 (i =1, . . . , 5) since there is no particular component which should be emphasized (we

could perhaps take α3 = 0 since there will be no v2 motion induced aligned with the

sheet). The MOL solution employed 350 grid points. In summary

γ 

=1.4,

¯B1

≡0

ρ|t =0 =

10−3 for x ∈ [0, 0.45]1 for x ∈ [0.45, 0.55]10−3 for x ∈ [0.55, 1]

m1|t =0 = m2|t =0 = 0

B2|t =0 =

1.1 for x ∈ [0, 0.45]0.6 for x ∈ [0.45, 0.55]1.0 for x

∈ [0.55, 1

]e|t =0 =

0.60525 for x ∈ [0, 0.45]0.98025 for x ∈ [0.45, 0.55]0.50025 for x ∈ [0.55, 1]

Homogeneous Neumann boundary conditions hold for all components, except for

momentum in the x-direction for which m1|∂ = 0.

In Figure 4.5 the density at time t  = 2, and the grid history until that time is shown.

The density panel again compares the adaptive solution with a reference result (with

Courant number 0.8), and it can be seen that the solution is somewhat influenced by

the higher (artificial) diffusion imposed. From the grid history, we conclude that the

timeframe shown is a little over two “periods” of the induced oscillation, which is

in agreement with the estimated period 0.97 as listed in [15]. Note how the MOL

technique nicely succeeds in tracing the waving motion of the sheet boundaries.

In contrast with the previous example, the adaptive method is able to maintain the

positivity of the thermal pressure for this case.

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FIGURE 4.5

Density at t  = 2 and grid history until that time for the oscillating plasma sheet.In the left panel, the MOL solution (solid) is again compared with a 1000-grid

point reference solution.

4.4 Towards 2D MHD Modeling

4.4.1 2D Magnetic Field EvolutionIn contrast to the 1D MHD case from above, multi-dimensional MHD simulations

face a non-trivial task when advancing a magnetic field configuration forward in time

while ensuring the property ∇ · B = 0. The core problem is represented by the

induction equation (4.4), alternatively written as

∂B

∂t = ∇ × (v × B) + mB (4.24)

with m the resistivity m ≥ 0. In two space dimensions, setting B = (B1, B2, 0), weobtain the following system of PDEs,

∂B1

∂t = mB1 + v1

∂B2

∂y− v2

∂B1

∂y+ B2

∂v1

∂y− B1

∂v2

∂y, (4.25)

∂B2

∂t = mB2 − v1

∂B2

∂x+ v2

∂B1

∂x− B2

∂v1

∂x+ B1

∂v2

∂x, (4.26)

together with the property ∇ · B = 0. This system will be solved using a 2D adaptive

grid method in Section 4.4.2.3.3, with particular attention paid to the solenoidal

condition.

One way to ensure a divergence-free magnetic field at all times is to make use of 

a vector potential formulation where B := ∇ × A. In two-dimensional applications,

the system (4.25) and (4.26) is then equivalent to the single PDE for the scalar A3

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component

∂A3

∂t = −v · ∇ A3 + mA3 , (4.27)

with∂A3∂y

= B1, − ∂A3∂x

= B2, while A = (0, 0, A3). We will use this simpler model

in Section 4.4.2.3.1 to compare different means for generating a 2D adaptive grid.

Note that magnetic field lines are isolines of this A3 potential.

Finally, we point out (cfr. [17]) that the partial problem posed by the system (4.25)

and (4.26), or equivalently the PDE (4.27), can be relevant as a physical solution to

the special case where we consider incompressible flow ∇ · v = 0, the momentum

equation (4.2) under the condition that the magnetic energy B2/2 is much smaller than

the kinetic energy ρv2/2, and the induction equation itself. In those circumstances,

the momentum balance decouples from the magnetic field evolution. In the modelproblems studied, we therefore impose an incompressible flow field v(x,y).

4.4.2 Adaptive Grids in Two Space Dimensions

4.4.2.1 Transformation in 2D

As in the 1D case we first make use of a transformation of variables

ξ 

=ξ (x , y , t ), η

=η(x , y , t ), θ  

=t , (4.28)

that yields for the Equation (4.27) (a similar derivation can be made for the (B1, B2)

system)

J A3,θ  + A3,ξ (xηyθ  − xθ yη) + A3,η(xθ yξ  − xξ yθ )

= A3,ξ 

−v1yη + v2xη

+ A3,η

v1yξ  − v2xξ 

+m

x2η + y2

η

J A3,ξ 

ξ 

−

xξ xη + yξ yη

J A3,η

ξ 

−

xξ xη + yξ yη

J A3,ξ 

η

+

x2ξ  + y2

ξ 

J A3,η

η

. (4.29)

Here, J  = xξ yη − xηyξ  is the determinant of the Jacobian of the transformation. In

general, we then allow for truly two-dimensionally deforming grids.

If we restrict the grid adaptation in a 1.5D manner, i.e., when we impose the extra

restriction xη = yξ  = 0, we get J  = xξ yη, and Equation (4.29) simplifies to

xξ yηA3,θ  − A3,ξ yηxθ  − A3,ηxξ yθ  = −v1yηA3,ξ  − v2xξ A3,η

+m

yηA3,ξ 

xξ 

ξ 

+

xξ A3,η

yη

η

. (4.30)

This dimensionally split approach for the grid adaptation will be compared with fully

2D deformations for the model problem from Section 4.4.2.3.1.

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4.4.2.2 Adaptive Grid PDEs in 2D

Due to the 2D transformation, now two fourth-order PDEs are needed to define

the grid and thereby the transformation. As an immediate extension of the 1D case

(4.11) we set S 1

xξ 

+ τ xξ θ 

W 1

ξ 

= 0 , (4.31)

S 2

yη

+ τyηθ 

W 2

η

= 0 ,

with S 1 and S 2 direction-specific versions of the operator S  defined in (4.12). The

weight functions in (4.31) are now

W 1 =  1 + α A23,x , W 2 =  

1 + α A23,y , (4.32)

for a fully 2D adaptive grid, while in the 1.5D case, we set

W 1 = 

1 + α maxy A23,x , W 2 =

 1 + α maxx A2

3,y . (4.33)

It can be shown (using 1D arguments in two directions), that with the latter choice

J 

=xξ yη > 0,

∀θ 

≥0 , (4.34)

so that this restricted grid adaptivity maintains the desirable property that grid cells

do not fold over. For the more general case (4.32), no guarantee can be given that

grid points will not collide! This could be called the “battle between adaptivity and

regularity.” The method parameters τ , σ , α are chosen in a similar way as before

and are specified per problem in the following sections. After semi-discretization of 

(4.30) and (4.31)we end upwith a banded ODE system with bandwidth = 6∗npts+2,

where npts × npts denotes the total number of gridpoints in 2D. This ODE system

is again time-integrated with DASSL [10].

4.4.2.3 Numerical Results

4.4.2.3.1 Kinematic Flux Expulsion

Thismodel problem dates back to1966[17], as one of the first studies toaddress the

role of the magnetic field in a convecting plasma. Starting from a uniform magnetic

field, its distortion by cellular convection patterns was simulated numerically for

various values of the resistivity m. We use this model problem to compare the 2D

with the 1.5D approach for two-dimensional moving grids.

Our 2D kinematic flux expulsion uses an imposed four-cell convection pattern withits incompressible velocity field given by

v(x,y) = (sin(2π x) cos(2πy), − cos(2π x) sin(2πy)) .

We solve for the scalar vector potential A3 from (4.27) on the domain (x,y) ∈[0, 1] × [0, 1] and for times t  ∈ [0, 0.5]. We set the adaptivity parameter α to unity,

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and τ  = 10−3, σ  = 1. The grid dimension is 25 × 25, while the resistivity is set

equal to m = 0.005. In terms of A3, the initial uniform vertical field is obtained

through A3|t =0 = 1 − x, while the boundary conditions are

A3|x=0 = 1, A3|x=1 = 0, A3|y=0 = A3|y=1 .

In Figure 4.6, we compare the grids and the obtained solution A3(x,y) at time t  = 0.5

for three simulation results. The top row uses a full 2D grid deformation, the middle

row takes the 1.5D adaptivity approach, while the bottom row shows a reference VAC

solution on a 100 × 100 uniform, static grid. The solution is shown both as a surface

and a contour plot, with the contour values varying between 0 and 1 with steps of 

0.05. Note that the 1.5D deformation works well for this case, since the steep parts

of the solution mostly vary in the x-direction. Although the 2D grid shows slightlysharper contour lines in the middle of the domain, the 2D deformation may break 

down at some point in time. For both cases we gain a factor of 16 in the total number

of grid points compared with the reference solution.

4.4.2.3.2 Advection of a Current-Carrying Cylinder

To demonstrate the dimensionally split grid adaptation on a case where truly 2D

deformations are required, we solve for the circular advection of a current-carrying

cylinder (taken from [13]).With a computational domain of size (x,y) ∈ [−50, 50]× [−50, 50], we embed

an isolated magnetic “flux tube” in a circulatory flow. The cylinder is specified by

A3|t =0 =

R/2 − [(x − x0)2 + (y − y0)2]/2R if (x − x0)2 + (y − y0)2 < R2,

0 elsewhere ,

and is initially centered at (x0, y0) = (0, 25) with radius R = 15 and the magnetic

field strength increases radially from zero to one at the cylinder edge. In terms of a

current J = ∇ × B, the cylinder has a constant axial current throughout.We simply rotate this current-carrying cylinder around (counterclockwise) by im-

posing

v(x,y) = (−y, x) .

When we solve for times t  ∈ [0, 2π], we then follow one period of revolution of 

the cylinder, at which time the initial configuration must be regained. The method

parameters are: adaptivity parameter α = 200 (due to the scaling-effect), τ  = 10−3,

σ =

1. We now use the dimensionally decoupled adaptivity on a 25×

25 grid

and a dimensionless artificial diffusion of 0.5 × 10−4. Boundary conditions do not

play a role in this example, so we simply took homogeneous Dirichlet conditions

A3|∂ = 0 everywhere. In Figure 4.7 we see the grids, solutions, and contour plots

at t  = 2π . The adaptive grid is nicely situated around the cylinder, although the

solution is slightly smoothed by the artificial diffusion term, which can also be seen

in the contour plot.

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FIGURE 4.6

Three solution strategies for 2D kinematic flux expulsion compared: each rowshows for time t  = 0.5 the grid, the solution for the vector potential A3(x,y) as

a surface plot, and as a contour plot (showing magnetic field lines) with fixed

contour levels at A3 = 0 : 0.05 : 1. The top row uses a 25 × 25 two-dimensionally

deforming grid, the middle row uses restricted 1.5D adaptivity, and the bottom

row is a 100 × 100 reference solution. The imposed velocity field is depicted at

bottom left.

4.4.2.3.3 Conservation of ∇ · B = 0?

To investigate how theadaptive method copes with the important property∇·B = 0,

we now take the full (B1, B2) system given by (4.25) and (4.26). Note that the current-

carrying-cylinder model is not appropriate for this purpose, since the initial condition

for the (B1, B2) system consists of piecewise linear parts (this follows from the initial

condition for A3 and B : = ∇ × A). As a consequence, constant weight functions W 1

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FIGURE 4.7

For initial time t =

0 (top row) and after one rotation at time t =

2π (bottom

row): Grid, surface plot of the potential A3, and magnetic field lines for the

current-carrying cylinder model with fixed contour levels at A3 = 0 : 0.325 : 7.5.

The top left frame shows the imposed circulation as a vector field.

and W 2 are obtained and therefore a uniform grid for all t  ≥ 0, independent of the

choice of the adaptivity parameter α. For this reason, we examine the (B1, B2) version

of the model in Section 4.4.2.3.1 with initial conditions B1

|t 

=0

=0, B2

|t 

=0

=1

and periodic boundary conditions. For simplicity we take the 1.5D approach. In

Figure 4.8, we show a plot of ∇ · B = 0 on a 30 × 30 adaptive grid at t  = 0.1, as

evaluated from a central difference discretization:

[∇ · B = 0]i,j  ≈B1,i+1,j  − B1,i−1,j 

xi+1 − xi−1

+ B2,i,j +1 − B2,i,j −1

yj +1 − yj −1.

Numerical values of ||∇·

B||∞

for different grid sizes are: 0.2117 (on a 20×

20 grid),

0.2104 (25 × 25 grid), and 0.1743 (30 × 30 grid). The main conclusion from these

results is that, although the grid concentrates near areas of high-spatial activity, the

solenoidal condition on the magnetic field is not preserved satisfactorily at all. This

is a severe drawback of the current MOL implementation. A possible remedy for this

could be adding a projection scheme after every time step, i.e., applying a Poisson

solver to correct the divergence of the magnetic field [1].

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FIGURE 4.8The divergence of the magnetic field for the solution in (4.4.2.3.3) at time t  =0.1 on a 30 × 30 adaptive grid. We evaluated the divergence from a centered

difference formula.

4.5 Conclusions

In this chapter we applied the adaptive MOL technique to various 1D MHD prob-

lems and 2D magnetic field evolution simulations. In 1D, accurate numerical results

were obtained for three important test cases. The method could further benefit from

specific MHD properties that have not been exploited in the present implementation.

For the 2D case, the adaptive method with restricted grid motion performed compa-

rably to fully 2D adaptive simulations. This is of interest for easier generalizations

to 3D calculations. Future work will consist of fully 2D MHD simulations and 3D

applications (model problems could be taken from [6, 9]). From our results, it is clear

that attention should be paid to means of maintaining pressure positivity in very low

pressure situations, more physically based artificial diffusion terms, and an appro-

priate remedy for ensuring the solenoidal condition on the magnetic field vector in

combination with the adaptive grid method for multi-dimensional applications. To

allow for the latter applications, we will switch to the use of iterative methods for the

linear systems behind the Newton process in the stiff ODE solver.

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Acknowledgments

RKperformed his workas partof the researchprogram of the association agreement

of Euratom and the “Stichting voor Fundamenteel Onderzoek der Materie” (FOM)

with financial support from the “Nederlandse Organisatie voor Wetenschappelijk On-

derzoek” (NWO) and Euratom. This work was partly performed in the project on

“Parallel Computational Magneto-Fluid Dynamics,” funded by the Dutch Science

Foundation (NWO) Priority Program on Massively Parallel Computing.

References

[1] J.U. Brackbill and D.C. Barnes, The effect of nonzero ∇ · B on the numerical

solution of the magnetohydrodynamic equations, J. Comput. Phys., 35, (1980),

426–430.

[2] M. Brio and C.C. Wu, An upwind difference scheme for the equations of idealmagnetohydrodynamics, J. Comput. Phys., 75, (1988), 400–422.

[3] E.A.Dorfi and L.O.’Drury, Simple adaptivegrids for 1-D initial valueproblems,

 J. Comput. Phys., 69, (1987), 175–195.

[4] H. Friedel, R. Grauer, and C. Marliani, Adaptive mesh refinement for singu-

lar current sheets in incompressible magnetohydrodynamic Flows, J. Comput.

Phys., 134, (1997), 190–198.

[5] W. Huang and R.D. Russell, Analysis of moving mesh partial differential equa-

tions with spatial smoothing, Research Report, 93-17, (1993), Mathematics and

Statistics, Simon Fraser University, Burnaby, British Columbia.

[6] R. Keppens, G. Tóth, R.H.J. Westermann, and J.P. Goedbloed, Growth and

saturation of the Kelvin-Helmholtz instability with parallel and antiparallel

magnetic fields, J. Plasma Phys., 61, (1999), 1–19.

[7] R. Keppens, M. Nool, P.A. Zegeling, and J.P. Goedbloed, Dynamic grid adap-

tation for computational magnetohydrodynamics, Lecture Notes in Computer Science, 1823, (2000), Springer Verlag, Berlin, 61–70.

[8] R.L. LeVeque, D. Mihalas, E.A. Dorfi, and E. Müller, Computational methods

for astrophysical fluid flow, Saas-Fee Advanced Course, 27, (1998), lecture

notes 1997, Swiss Society for Astroph. and Astron., O. Steiner and A. Gautschy,

eds., Springer Verlag, Berlin.

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[9] M.G. Linton, G.H. Fisher, R.B. Dahlburg, and Y. Fan, Relationship of the mul-

timode kink instability to δ-spot formation, Astrophys. J., 522, (1999), 1190–

1205.

[10] L.R. Petzold, A description of DASSL: A differential/algebraic system solver, IMACS Transactions on Scientific Computation, (1983), R.S. Stepleman et al.,

eds., North-Holland, Amsterdam, 65–68.

[11] W.E. Schiesser, The Numerical Method of Lines, Integration of Partial Differ-

ential Equations, Academic Press, (1991), San Diego, CA.

[12] J.M. Stone and M.L. Norman, ZEUS-2D: a radiation magnetohydrodynamics

code for astrophysical flows in two space dimensions. II. The magnetohydro-

dynamic algorithms and tests, Astrophys. J. Suppl., 80, (1992), 791–818.

[13] G. Tóth and D. Odstrcil, Comparison of some flux corrected transport and total

variation diminishing numerical schemes for hydrodynamic and magnetohy-

drodynamic problems, J. Comput. Phys., 128, (1996), 82–100.

[14] G. Tóth, A general code for modeling MHD flows on parallel computers: Ver-

satile Advection Code, Astrophys. Lett. & Comm., 34, (1996), 245–250.

[15] G. Tóth , R. Keppens, and M.A. Botchev, Implicit and semi-implicit schemes

in the Versatile Advection Code: numerical tests, Astron. & Astroph., 332,

1159–1170.

[16] R.A. Trompert, Local uniform grid refinement for time-dependent partial dif-

ferential equations, CWI-tract, 107, (1995), Centrum voor Wiskunde en Infor-

matica, Amsterdam.

[17] N.O. Weiss, The expulsion of magnetic flux by eddies, Proc. of Roy. Soc. A,

293, (1996), 310–328.

[18] P.A. Zegeling, Moving grid methods for time-dependent partial differential

equations, CWI-tract, 94, (1993), Centrum voor Wiskunde en Informatica, Am-sterdam.

[19] P.A. Zegeling, r-refinement for evolutionary PDEs with finite elements or finite

differences, Appl. Num. Maths., 26, (1998), 97–104.

[20] U. Ziegler, A three-dimensional Cartesian adaptive mesh code for compressible

magnetohydrodynamics, Comp. Phys. Comm., 116, (1999), 65–77.

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Chapter 5

 Development of a 1-D Error-Minimizing Moving Adaptive Grid Method 

Mart Borsboom

Abstract Instead of developing an adaptive grid technique for some discretization

method, we develop a discretization technique designed for grid adaptation. This

so-called compatible scheme allows to translate the leading term of the local residual

directly in terms of a local error in the numerical solution (the numerical modeling

error). An error-dependent smoothing technique is used to ensure that higher-order

error terms are negligible. The numerical modeling error is minimized by meansof grid adaptation. Fully converged adapted grids with strong local refinements are

obtained for a steady-state shallow-water application with a hydraulic jump. An un-

steady application confirms the importance of taking the error in time into account

when adapting the grid in space. We discuss the shortcomings of the present imple-

mentation and the remedies currently under development.

5.1 Introduction

In moving adaptivegrid methods, both the numerical solution and the grid on which

that numerical solution is defined are considered unknown. This offers an enormous

increase in numerical modeling flexibility, but also raises the far from easy question

of how to couple the grid to the numerical solution. The standard answer is to apply

an equidistribution principle: the grid is defined by the equidistribution of a solution-

dependent error measure or monitor function over the grid cells. Many differenterror measures have been proposed in the literature, often chosen heuristically, with

little or no justification [12]. Ideally, error measures include all important sources of 

numerical solution errors, i.e., terms that indicate how grid resolution, grid stretching,

grid curvature, and grid skewness affect the accuracy of the numerical simulation.

If essential error information is missing or not used properly, grid adaptation may

not give any improvement [27]. The use of an incomplete or incorrect error estimate

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could even lead to an increase of solution errors due to the incorrect grid point dis-

tributions and/or the severe grid distortions that it may induce. This can be avoided

by adding terms that ensure some form of grid regularity or that limit grid variation,

thereby reducing at the same time the adaptability of the grid. The resulting adaptivegrid technique will probably require problem-dependent fine tuning as well. These

arguments clearly illustrate the importance of a reliable error measure when designing

a moving adaptive grid algorithm [12, 17].

Residual-based, a posteriori error estimates for hyperbolic problems of practical

interest (inparticular nonlinear flow problems) still lack sharpness or arenotgenerally

applicable [11, 15, 17]. Using the residual as such is generally not a good idea. The

model equations (and hence the residual) can be multiplied by any smooth positive

function without changing the solution. Depending on this scaling, a large/small

residual may therefore not correspond with a large/small solution error. The relationbetween the solution error and the residual can be estimated by considering a global

dual problem, obtained after linearization, which is usually complex and expensive

to solve. Solving a dual problem locally is practically more feasible [23], although

it does indicate only the locally generated error and ignores solution errors that are

the result of the accumulation during propagation of errors generated elsewhere [14].

However, a small local residual may be due to the cancellation of numerical errors

originating from different modeling terms and may therefore not correspond with

small errors as such. It is usually also not clear how the local residual depends on

the local grid parameters (size, stretching, curvature, skewness), which makes this

approach less suited for error-minimizing grid adaptation. Furthermore, it may be

difficult to develop a meaningful local dual problem with appropriate local (inflow,

outflow) boundary conditions.

Propagation, andhence error propagation, is typical of flow andtransport problems;

it makes the development of an error-minimizing adaptive grid technique for such

applications extremely complicated because of the obscure and complex relation that

exists between the regions where the solution errors are generated and the regions

where the solutionerrorsmanifest themselves. This applies to numerical errorsas wellas to physical errors and errors due to incorrect data. Examples of physical modeling

errors are the approximate description of turbulence or the neglect of certain aspects

like a space dimension or viscous effects. Data errors may consist of uncertainties

in, e.g., the initial and boundary conditions, geometry, and certain model parameters.

When reducing numerical errors through grid adaptation, the presence of these other

modeling errors should be taken into consideration as well [7]. In particular, refining

the grid is not useful in regions where the solution error is mainly due to the applied

physical model or caused by unreliable or incomplete input data. This strongly limits

the usefulness of an adaptive grid technique that merely attempts to minimize thenumerical solution error. In complex applications it is, however, virtually impossible

to quantify the effect of physical modeling errors and data errors on the solution, let

alone to take the relative importance of this effect into account in the grid adaptation.

We have, therefore, not opted for the development of a grid adaptation method that

aims to reduce some upper error bound to a given tolerance level. Instead, our goal

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is to minimize the part of the numerical solution error that is generated locally as a

result of using a numerical approximation of the model equations. We will refer to

this error as the numerical modeling error. Note that it may be feasible to compare

the numerical modeling error with physical modeling errors and data errors. Thiscould then be used to assess its relative importance and hence the usefulness of local

(adaptive) grid refinements.

The basis ofour method is the approximation of the local residual in terms of a local

solution error. We want to avoid the use of the residual as such in combination with

a local dual problem because of the disadvantages mentioned before. Investigating

this issue more closely, we found that it is generally impossible to reformulate the

residual directly in terms of local errors in the numerical solution. The reason for

this seems to be the discrepancy that often exists between the discretization of the

model equations and the way the numerical solution is represented. For example,the use of piecewise linear polynomials to reconstruct the numerical solution over

the whole domain is consistent with the use of a second-order accurate discretization

technique. This does not mean however that the second-order interpolation error is

representative of the numerical modeling errors.

So instead of analyzing the residual of some numerical scheme, we decided to

investigate numerical schemes for their suitability of rewriting the residual as a local

solution error. This has led to the development of a discretization technique designed

for error analysis and hence for grid adaptation. Using this technique, the residual of any flow or transport equation discretized on a non-uniform, moving grid can indeed

be reformulated, at least in 1-D, in terms of a local solution error.

The moving adaptive grid equations are obtained by solving an optimization prob-

lem, minimizing the numerical modeling error in the L1 norm. The L1 norm has been

selected because of its physical relevancy (see also [29]). In particular, the numerical

approximation of a discontinuity spread over a fixed number of grid points shows

only the expected first-order error behavior if it is measured in the L1 norm. This is

consistent with the local order of accuracy of the numerical scheme.

5.2 Two-Step Numerical Modeling

In the numerical error analysis that we will present we will make extensive use

of truncated Taylor-series expansions. A smoothing technique is applied prior to the

discretization to ensure that higher-order error terms are negligible. This makes thenumerical modeling process essentially a two-step procedure. In the first step, the

problem to be solved is regularized by adding a suitable form of smoothing; in the

second step, the regularized problem with smooth solution is discretized. The amount

of smoothing in the first step is controlled by the error that is made in the second step

by an error feedback loop. Smoothing or regularization is used frequently in adaptive

grid techniques [5, 8, 12, 17].

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Figure 5.1 shows the elements of the two-step method. We discern a smoothing

step and a discretization step connecting four different problems:

• the difficult problem, i.e., the originalphysicalmodel problem, whose solution

may include steep gradients and discontinuities

• the easy problem, i.e., the regularized problem, whose solution is smooth

enough to be discretized with sufficient accuracy

• the discretized problem, obtained upon the discretization of the easy problem

• the equivalent problem, i.e., the differential problem equivalent with the dis-

cretized problem

FIGURE 5.1

Outline of the two-step numerical modeling technique.

The smoothing step converts the difficult problem into the easy problem by explicitly

adding suitable smoothing terms. The discretization step converts the easy problem

into the equivalent problem by means of a suitable discretization of the easy problem.

To obtain the second conversion in an explicit form, we determine the residual asso-ciated with the discretized problem. The residual can be viewed as the continuous

equivalent of the discretization error.

The difficult, easy, and equivalent problem are all differential problems. They each

consist of a system of partial differential equations (L(u) = 0, L(u) = 0, L(u) = 0)

and a set of data (data,data,   data) containing, e.g., the boundary and initial condi-

tions. Functionsand parameters required to fully specify each problem (geometry and

source terms, for example) are also included in the data. The discretized problem, on

the other hand, consists of a system of algebraic equations (L(u)

=0) supplemented

with a discrete data set (data).

The discretized problem is usually the only problem that is solvable. However, we

do not know the differential problem corresponding with its solution u. In contrast,

the difficult and easy problem are fully specified but their solution (respectively, u

and u) is largely unknown. The link between both is the equivalent problem. The

equivalent solution u and data   data are a close and smooth approximation of the

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solution and data of the discretized problem, while the equivalent system of equations

L(

u) = 0 is an approximation of the system of partial differential equations satisfied

byu given   data. If the discretization is consistent, the equivalent problem is also an

approximation of the easy problem that in turn is an approximation of the difficultproblem.

Some form of two-step modeling is used in virtually any discretization method for

flow and transport problems of practical interest. Smoothing may have been added

explicitly (i.e., prior to the discretization) in the form of artificial dissipation terms

in the equations, or implicitly by incorporating a dissipative mechanism like flux

limiters inside the discretization method. Usually, the purpose of adding dissipation

is to reduce wiggles or to ensure monotonicity. Here we demand that it guarantees a

sufficiently smooth solution, i.e., negligibly small higher-order discretization errors,to allow a meaningful error analysis that can be used in a grid adaptation procedure.

A simple example may illustrate the connection between smoothing and grid adap-

tation. We consider a smooth function with a discontinuity at x = 0 and approximate

this function by means of a continuous piecewise linear discrete function defined on

the grid x = ±1, ±3, . . . . Because the discontinuity is at the center of a grid cell, its

best possible numerical approximation is only one grid cell wide. We now discretize

the function on the grid x = 0, ±1, ±2, . . . and observe that there is no improvement

in accuracy, despite the fact that the resolution has been doubled. The reason forthis is obvious: the discontinuity is now at a grid point and must be spread over two

grid cells. It is even possible to get worse results on a finer grid, simply because the

position of the discontinuity changes from a cell center to a grid point (consider the

grid x = 0, ±1.5, ±3, . . . ). Such a function has been considered in [2]. The results

presented in that paper show that although the global error behavior in the L1 norm is

indeed first order, without smoothing of the discontinuity the approximation error is

an irregular function of the number of grid points. When the function is first properly

smoothed, the L1 approximation error decreases uniformly as the number of grid

points increases, showing a clear first-order behavior independent of the position of the grid points.

The dependence of the error on grid point position is unacceptable in the context

of the use of moving adaptive grid techniques where we need a monotone relation

between the numerical accuracy and the grid resolution. The example shows that the

behaviorofhigher-ordererrorcomponents canbe rathererratic andbecomes dominant

if the solution is not smooth. This can be avoided by smoothing the function that is to

be approximated (e.g., the solution of a differential problem) over a sufficient number

of grid cells. It can be easily verified that the smoother the function, the less sensitiveits numerical approximation is to the position of the grid points. However, smoothing

introduces another form of numerical error, so in practice a compromise needs to

be found between the amount of smoothing (should be as small as possible) and the

amount of spreading (must be large enough). In particular, no additional smoothing

is required if the function is already sufficiently smooth. These requirements are

precisely the design criteria of the two-step method.

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5.3 1-D Shallow-Water Equations

The physical problem that we consider is the modeling of free-surface flow through

an open channel section. In many practical applications, the vertical and transversal

length scales are small compared to the longitudinal scales and can be neglected.

Assuming constant density ρ and restricting ourselves to channels of constant width

W , the flow can then be modeled by the 1-D equations (see, e.g., [3]):

∂d 

∂t + ∂q

∂x= 0 , (5.1)

∂q

∂t + ∂q2/d 

∂x+ gd 

∂h

∂x+ g

P q|q|W d 2C2

= ∂

∂x

νartd 

∂q/d 

∂x

. (5.2)

Space coordinate x [m] is defined along the axis of the channel. The unknowns of 

the flow equations are the water depth d  [m] and the depth-integrated flow velocity

in x-direction q [m2/sec]. Depth-averaged flow velocity u [m/sec] is equal to q/d .

Gravitational acceleration g [m/sec2] is a given constant parameter. Chézy coefficient

C

[m1/2/sec

]is a friction parameter to account for friction losses due to the bottom

friction. Water level h [m] is the sum of d  and the given bottom level of the channelzb [m]:

h = d + zb , (5.3)

while wetted perimeter P  [m] is defined as P  = W  + 2d .

Continuity equation (5.1) is a mass conservation law. It describes the balance

between the rate of change of mass in a cross-section and the net mass flow entering

that cross-section for constant ρ and W . The terms in the left-hand side of momentum

equation (5.2) represent the rate of change of momentum, the net momentum flow, thehydrostatic pressure force, and the bottom friction force, respectively. In the right-

hand side of (5.2) we have the added artificial smoothing term, so (5.1) and (5.2)

form in fact the equations of an easy problem (cf. Figure 5.1). Since we will consider

the easy problem only, we have omitted the tilde for convenience. The form of the

smoothing term is equal to the physical viscosity term integrated over the water depth,

neglecting non-conservative parts to avoid artificial loss of momentum. Its viscosity

coefficient νart [m2/sec] is of course artificial, although the same formulation could

also be used to improve the modeling of physical effects [18].

Equations (5.1) and (5.2) can be recombined to the characteristic equations:

(c ∓ u) cont. eq. (5.1) ± mom. eq. (5.2) , (5.4)

with c = √ gd  the wave celerity. These equations describe the propagation of the

Riemann variables (c∓u)∂d ±∂q along the characteristics dx/dt  = u ± c. The flow

behavior depends to a large extent on the direction of the characteristics. Introducing

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the FroudenumberFr = |u|/c we make a distinctionbetween subcriticalflow (Fr < 1;

propagation in both directions), critical flow (Fr = 1; one propagation direction

vanishes), and supercritical flow (Fr > 1; only propagation in downstream direction).

Note that in the shallow-water model there will always be some information movingupstream (even if Fr > 1), due to the diffusive transport introduced by the artificial

viscosity. This is perfectly acceptable if the effect is small enough.

In regions where the solution of (5.1) and (5.2) is differentiable, the equations can

be recombined to the energy equation:

∂

∂t 

d 

1

2u2 + gh − 

1

2gd 

+ ∂

∂x

ud 

1

2u2 + gh

= −g

P u2

|u

|W C2 − νartd ∂u

∂x2

+∂

∂x uνartd 

∂u

∂x . (5.5)

Ignoring the dissipation terms in the right-hand side, the steady-state solution of 

Equation (5.5) reads (from (5.1) we obtain ud  = q = constant):

h + u2

2g= constant . (5.6)

This Bernoulli equation shows that the energy head, h

+ 12

u2/g, is constant in smooth

stationary frictionless flow; no energy is dissipated. The equation is not valid across asteady discontinuous hydraulic jump where we have to apply the Rankine-Hugoniot

relations obtained from (5.1) and (5.2):

(ud )1 = (ud )2 ,u2d + g

d 2

2

1

=

u2d + gd 2

2

2

. (5.7)

The indices 1 and 2 indicate the state left and right of the discontinuity. Solving (5.7)

across a hydraulic jump gives the energy loss across the jump. Further details can befound in [4]. Using the steady-state solution of (5.1), discharge q = ud  = constant,

the solution of (5.6) (in regions where the solution is smooth) and (5.7) (across a

hydraulic jump) can be determined analytically, given suitable boundary conditions.

This provides a useful analytical bench-mark solution to compare numerical solutions

with [10, 21].

We point out that the discontinuous hydraulic jump is only a solution of (5.1) and

(5.2) if the artificial viscosity is set tozero, i.e., if the model equations are considered to

form a difficult problem (cf. Figure 5.1). With the addition of the artificial smoothing

term, it has become an easy problem; discontinuities are spread and “replaced” so

to speak by gradients of limited steepness. It is easily verified that the spreading is

proportional to the value of νart.

At some distance of a smooth but nevertheless steep gradient, the solution gradients

will be small again, and the effect of the artificial viscosity negligible. Neglecting

the variations in channel geometry, it follows from momentum equation (5.2) that

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the jump relations (5.7) (or the unsteady equivalent) are still satisfied across the

 jump region. So if only a moderate amount of smoothing is applied, the behavior

of discontinuities is still modeled accurately. The condition to be satisfied here is

that within a smoothing region the channel variation must be small. This conditionwas formulated 50 years ago by Von Neumann and Richtmyer: “ . . . for the assumed

form of dissipation, and for many others as well, the Rankine–Hugoniot equations

are satisfied provided the thickness of the shock layers is small in comparison with

other physically relevant dimensions of the system” [26].

An interesting aspect of the easy problem is that because the solution is smooth and

differentiable, energy equation (5.5) is valideverywhere, also across the jump regions.

In fact, the sudden energy drop across the jump is replaced by a steep decrease of the

energy head due to the artificial viscosity. Since jump conditions (5.7) are still valid,

the net result must be the same.

White presents an analysis of the structure of a 1-D viscous aerodynamic shock 

wave [28]. The analysis can be applied to hydrodynamic shocks as well. His results

confirm that the thickness of the shock is proportional to the size of the viscosity

coefficient. A rather surprising result is the entropy overshoot in the shock which is

due to the energy redistribution caused by the viscous term. In the inviscid limit, the

overshoot becomes a peak ofzero thicknessandvanishes, leaving only thewell-known

discontinuous increase of entropy. In our model the entropy overshoot corresponds

with an undershoot of the energy head. The last term in the right-hand side of (5.5)is responsible for this effect. It is first negative (u decreases, hence ∂u/∂x becomes

strongly negative inside the shock), but becomes positive at the end of a viscous shock 

layer. Overall the third term does not affect the energy across a shock because of its

conservative form. The energy across a shock decreases because of the negative-

definite second term in the right-hand side that also prevents the development of non-

physical expansion shocks (the entropy condition, cf. [13]). At the end of a shock,

however, where the third term is strongly positive and dominant, the right-hand side

of (5.5) becomes positive causing a small energy uplift. As explained before, the

overall energy loss across a viscous shock can still be a very close approximation of the inviscid shock loss.

The conclusion that can be drawn from this discussion is that although the solutions

aredifferent locally, globally thesolution of theeasy problem andthedifficult problem

may be expected to be virtually the same provided that not too much artificial viscosity

is added. This is precisely the purpose of grid adaptation; the amount of artificial

dissipation required depends on both the solution and the grid, so by optimizing the

grid as a function of the solution, the artificial viscosity can be minimized and the

accuracy maximized. This too was perceived by Von Neumann and Richtmyer: “the

qualitative influence of these terms (read: the artificial viscosity) can be made as

small as one wishes by choice of a sufficiently fine mesh” [26].

A condition to be fulfilled by the numerical scheme is that the solution of the easy

flow problem is calculated with sufficient accuracy. To be able to capture all details

of the (artificially thickened) viscous shock, including the undershoot of the energy

head, shocks must be spread over at least several grid cells. It seems that this argument

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can be reversed: if a shock is not modeled as a discontinuity (any shock-capturing

method), it should be modeled as a viscous shock of a certain thickness in order to be

physically realistic. Upwind schemes that spread shocks over only one or two grid

cells may not be capable of modeling the entropy overshoot, and hence the dynamics,of that shock correctly. This observation is in line with the discussion of Section 5.2

and emphasizes the importance of sufficient resolution and hence smoothing.

5.4 Compatible Discretization

Our investigations have revealed that there seems to be only one way to obtain a

discretization of flow and transport equations with the property that the residual can

be formulated in terms of errors in the numerical solution and other variables. The key

element of this approach is the definition of unique approximations per grid cell of all

variables. We have used piecewise polynomial approximations that are defined on a

uniform grid in the parameter space or computational space ( ξ , τ ). It is convenient

for the discretization of the model equations and for the error analysis to define the

differential problem in this computational space as well.

Since we consider moving grids, the mapping of the computational space ( ξ , τ )onto the physical space (x,t) is defined by the two functions:

x = x(ξ, τ) , t  = t(τ) . (5.8)

Actually, it is defined by several such functions because each problem that we con-

sider in the two-step modeling technique (cf. Figure 5.1) requires its own coordinate

transformation. This is only relevant for the two problems related to the numerical

scheme: the discretized problem and the equivalent problem. We will see later that it

is indeed essential to consider for each of these two problems a separate coordinate

transformation.

The discretization in time that we will apply is a two-level method. As a conse-

quence, each time step can be considered separately, i.e., each time step the transfor-

mation in time can be redefined including a change of the size of the time step. This

permits us to restrict ourselves to transformation functions (5.8) that are linear in τ :

t τ 

=1 , xτ τ 

=0 , (5.9)

where we have used the convention that a subscript that is a coordinate denotes

differentiation with respect to that coordinate.

The (moving) space-time grid that we use is shown in Figure 5.2. The circles in

that figure indicate the coordinates (xni , t n) of the grid points (i, n) determine the

coordinate transformation of the discretized problem. Using (bi)linear interpolations

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per grid cell, the transformation is defined by:

x(ξ,τ)

=ξ i − ξ 

ξ  τ n − τ 

τ 

xn−1i−

1

+τ  − τ n−1

τ 

xni−

1+ ξ  − ξ i−1

ξ 

τ n − τ 

τ xn−1

i + τ  − τ n−1

τ ξ xn

i

,

t ( ξ , τ ) =τ n − τ 

τ t n−1 + τ  − τ n−1

τ t n , (5.10)

with:

ξ i−

1

≤ξ 

≤ξ i , i

=1, . . . , I  

+1 ,

τ n−1 ≤ τ  ≤ τ n , n = 1, . . . , N .

We have ξ i = ξ i−1 + ξ,i = 1, . . . , I  + 1, with ξ  the size of the uniform grid in

computational space. The grid consists of I  + 1 grid cells, with I  grid points inside

the domain and 2 virtual grid points outside the boundaries (cf. Figure 5.2). The

reason for this will become clear in Section 5.4.1. In order to satisfy (5.9), we take

τ n − τ n−1 = τ  = t  within each time step [τ n−1, τ n]. As explained before, this

does not preclude the use of different values of t  in different time steps. The total

number of time steps is N .

FIGURE 5.2Grid in physical time and space.

Using (5.8) and(5.9), Equations (5.1) and(5.2) transformed to computational space

can be written as:

∂(xξ d)

∂τ + ∂(q − xτ d)

∂ξ = 0 , (5.11)

∂(xξ q)

∂τ + ∂((q/d − xτ )q)

∂ξ + gd 

∂h

∂ξ + xξ g

P q|q|W d 2C2

= ∂

∂ξ 

νartd 

xξ 

∂q/d 

∂ξ 

.

(5.12)

This result has been obtained upon multiplying the transformed equations by xξ 

and rearranging the terms to recover the conservative form of the original system

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formulated in physical space. Notice that the coefficients t τ   have vanished because

of (5.9).

For the discretization of (5.11) and (5.12) we introduce uniquely defined finite-

dimensional function approximations of all variables: piecewise linear interpolationsin computational space but backward piecewise constant (and hence discontinuous)

extrapolations in computational time. The latter makes that the discretization in time

will have the form of a backward Euler scheme. It provides the amount of dissipation

in time required to ensure stability for any size of the time step, and also simplifies

the analysis of the error in time. The accuracy obtained with this first-order scheme

in time may still be quite acceptable, provided that we are able to design a moving

adaptive grid method capable of aligning the grid properly with the solution. So we

decided to use:

d(ξ,τ) = ξ i − ξ 

ξ d ni−1 + ξ  − ξ i−1

ξ d ni , (5.13)

ξ i−1 ≤ ξ  ≤ ξ i , i = 1, . . . , I  + 1 ,

τ n−1 < τ  ≤ τ n , n = 1, . . . , N .

Similar expressions are used for unknown q, wetted perimeter P , and artificial vis-

cosity coefficient νart. The expression is assumed to exist for discretized water level

h. As before, we have used the overbar to indicate discrete functions (cf. Figure 5.1).

 All discrete functions, i.e., x [Equation (5.10)], d  [Equation (5.13)], q, h, P , and

νart, must be (made) sufficiently smooth [the smoothness of t  is guaranteed because

of (5.9)]. This condition must be fulfilled in order that higher-order error terms can

be neglected in the error analysis. The smoothing step with error feedback loop (cf.

Figure 5.1) should take care of that. As for the unknowns d  and q, their smoothness

is realized by the artificial viscosity term. To ensure smoothness of the artificial

viscosity coefficient (an essential part of the method!), a separate equation is applied:

νart − αξ 2∂2νart

∂ξ 2= cν Err ν . (5.14)

Function Err ν is an error expression of dimension [m2/sec] that will be specified

later. For the moment it is sufficient to mention that Err ν (and hence νart) is O(x3)

in regions where the solution is smooth and O(x) in regions where steep gradients

develop. This makes that the artificial viscosity mechanism does not affect the formal

second-order accuracy of the scheme, while discontinuities will be spread over a fixed

number of grid cells depending on the value of constant scaling coefficient cν .

The amount of smoothing applied in (5.14) is determined by constant coefficient α.

The smoothing of νart is required to reduce the unreliable higher-order error infor-

mation that may be present in Err ν to an insignificant level. These errors are partly

due to the neglect of higher-order errors in the error analysis and partly introduced

by approximating Err ν by means of a discretization (details later). The smoothing

of νart is in computational space because Err ν will be determined and discretized

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in computational space. The use of a constant smoothing coefficient α is sufficient

because the artificial viscosity coefficient by itself is already a higher-order term.

An implicit smoothing equation like (5.14) is used frequently in moving adaptive

grid methods. To see this, replace νart by smoothed grid-point concentration n, α byα(1+α), and cν Err ν by non-smooth pointconcentration n, and discretize the equation

by means of finite differences. The result is the equation that Dorfi and Drury use to

ensure grid smoothness [5]. Huang and Russell show that smoothing of the monitor

function and of the grid point concentration are equivalent [16]. Implicit smoothing

of grid point concentration has been used in, e.g., [9, 25, 30, 31]. An explicit monitor

smoothing technique approximating implicit smoothing has been used in [20, 22].

We will use an equation like (5.14) also for the smoothing of the error expressions

that are used in the moving adaptive grid procedure, to eliminate higher-order errors

that are not included properly. This automatically takes care of grid smoothness, i.e.,no special measures are required to ensure that x is sufficiently smooth.

One variable may not be sufficiently smooth: h. This is presently one of the main

shortcomings of the method. Water level h is equal to the sum of bottom level zb

and water level d  [cf. Equation (5.3)] and considered as a function of d . It is obvious

that smoothness of d  is no guarantee for smoothness of h; that depends entirely on

the given profile zb that in practice may be highly irregular. Moreover, while d  is

a function that is piecewise linear in computational space and piecewise constant

in computational time, the behavior of h can be anything depending on how zb is

specified. Usually, zb is given as a piecewise linear function based on the available

data of a channel’s geometry. This does not make h a function that is linear per

grid cell because the coordinates where the geometry is specified will rarely coincide

with grid points. Discrete function h will generally not be piecewise constant in time

either, even though zb is constant in time. This is because zb is constant in time in

physical space, not in computational space.

All this is ignored at present. That is, also for h an expression like (5.13) is used,

but a rather rough approximation is used to define the grid point values hni :

hni = d ni + zb

x

n− 12

i

, (5.15)

with:

xn− 1

2

i = x

ξ i , τ n− 1

2

= 1

2

xn−1

i + xni

.

To obtain the bestpossibleaccuracyin timeweevaluate the bottomlevel at the position

of the moving grid in the middle of each time step.

5.4.1 Discretized Shallow-Water Equations

In the previous part we defined function approximations that are piecewise linear in

computational space and piecewise constant in computational time. It is reasonable to

assume that this is sufficient to construct a discretization that is second-order accurate

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in space and first-order accurate in time, in agreement with the leading interpolation

errors. However, the first derivative of a piecewise linear approximation is piecewise

constant, and only second-order accurate at cell centers. The second derivative of 

a piecewise linear function is not defined. As a result, (5.11) and (5.12) cannot bediscretized directly.

The obvious solution is to consider a weak formulation, integrating the model

equations in space from cell center to cell center which are the only positions where

the viscous fluxes can be evaluated with second-order accuracy. This automatically

defines a finite volume technique, with volumes as indicated by the solid lines in

Figure 5.2. Since all discrete functions have been specified, the discretization in

space is now fully defined and straightforward to obtain.

It is important to evaluate the integrals in space of the different terms, with the

functions replaced by their discrete approximations, with at least  fourth-order ac-curacy. Second-order accurate approximations of the integrals, obtained by using,

e.g., 1-point Gauss quadrature rules, are not allowed. Although that would lead to a

second-order accurate discretization in space as well, it wouldnot leadtoa compatible

discretization. It would introduce errors that are of the same order as the interpolation

errors, thereby effectively modifying the defined discrete functions. In consequence,

the interpolation error of these functions would not be the only source of discretization

errors anymore. To make sure that the second-order interpolation errors are included

with at least second-order accuracy, the integrals must be approximated fourth-order

accurate or better.We give two examples of how this compatible discretization in space works out

in practice, keeping everything in time continuous for the moment (method of lines

[MOL] approach). The time derivative of (5.12) is discretized in space according to: ξ i+ 1

2

ξ i− 1

2

∂(xξ q)

∂τ dξ  ≈

 ξ i+ 1

2

ξ i− 1

2

∂(xξ q)

∂τ dξ 

=∂

∂τ (xi

−xi

−1)

qi−1 + 3qi

8 +(xi

+1

−xi )

3qi + qi+1

8 , (5.16)

while the space discretization of the artificial viscosity term of (5.12) reads: ξ i+ 1

2

ξ i− 1

2

∂

νartd/xξ ∂(q/d)/∂ξ 

∂ξ dξ 

≈

νartd 

xξ 

∂q/d 

∂ξ 

i+ 1

2

−

νartd 

xξ 

∂q/d 

∂ξ 

i− 1

2

, (5.17)

with:νartd 

xξ 

∂q/d 

∂ξ 

i+ 1

2

=

νart

xξ 

∂q

∂ξ − νartq

xξ d 

∂d 

∂ξ 

i+ 1

2

= νart,i + νart,i+1

2

qi+1 − qi

xi+1 − xi

− qi + qi+1

d i + d i+1

d i+1 − d i

xi+1 − xi

.

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Note that this discretization has not been obtained by taking the average of νartd/xξ 

and the difference of q/d  evaluated at the grid points. That would imply that cer-

tain combinations of variables rather than the variables themselves are approximated

piecewise linearly. That is incompatible with the piecewise linear approximation of those variables used in the other terms [e.g., in time discretization (5.16)].

Our research has indicated that it is this mix of different variable approximations

often encountered in numerical discretizations that makes it impossible to express

discretization errors made in the equations in terms of errors in the numerical solution.

One would expect this error to be some interpolation error, but if several interpolation

errors have been mixed together during the discretization process, it is obviously not

possible to recover a well-defined one afterwards. See also [2].

There are still a few details to be filled in. One concerns the discretization of (5.14)

which reads:3

4+ 2α

νart,i +

1

8− α

νart,i−1 + νart,i+1

= cν

ξ 

 ξ i+ 1

2

ξ i− 1

2

 Err ν dξ . (5.18)

The discretization of the integral in the right-hand side is not critical; higher-order

errors are irrelevant in the determination of νart and are damped by the smoothing

anyway. The expression for Err ν will be given in Section 5.5.3.

The discretization in time is simply obtained by evaluating the equations at the

central time level τ n− 12 = 

12

(τ n−1 + τ n) using the discrete function approximations

in time. This is within O(τ 2) equal to the discretization obtained by integrating

the equations over one time step, which is sufficient for compatibility since the time

discretization is only first-order accurate.

As for the boundary conditions, they have to be applied at a cell center to allow the

whole domain to be covered with finite volumes (cf. Figure 5.2). This is rather fortu-

nate since both Dirichlet and Neumann boundary conditions can then be discretized

with second-order accuracy using only two grid points. Compatibility is no prob-

lem either. We will consider only subcritical flow at boundaries, and apply Dirichletconditions (either q or h imposed) supplemented with Equation (5.4) describing the

behavior of the outgoing characteristic. The discretization of the characteristic equa-

tion is obtained in two steps:

• The flow equations (5.11) and (5.12) are discretized at the boundaries using the

defined discrete approximation of the variables. Because the approximation

is linear in space, second derivatives vanish, but the artificial viscosity term is

still contributing to the discretization. To be able to discretize that term at the

boundary, we write it as:

∂

∂ξ 

νartd 

xξ 

∂q/d 

∂ξ 

=νart

xξ 

∂2q

∂ξ 2− q

d 

∂2d 

∂ξ 2

+ 1

xξ 

∂νart

∂ξ − νart

d 

∂d 

∂ξ 

∂q

∂ξ − q

d 

∂d 

∂ξ 

.

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underrelaxation coefficient of 0.8 and multiply the part of ∂L/∂u stemming from the

linearization of the artificial viscosity term by a factor 1.3. Without the latter, a much

stronger underrelaxation of νart is usually required. We found that the combination of 

both measures is sufficient to stabilize the implicit iterative solver while maintaininga good performance, despite the fact that νart, which can be very important locally, is

treated explicitly.

To ensure stability at the beginning of an iteration process and at the same time

obtain a high convergence speed, we have made the pseudo time step per grid point

inversely proportional to the local previous solution correction (um−1 − um−2). The

result is a pseudo time step that automatically compensates for the local Newton

linearization error. To reduce the sensitivity of the method to irregular conver-

gence behavior, the pseudo time steps are slightly underrelaxed. The iteration pro-

cess is initialized by taking the initial pseudo CFL number CF L0pseu = (|q0|/d 0 +

(gd 0

)1/2)t 0pseu/x equal to 1. This gives a very conservative initial pseudo time-

step value, which in fact is only required for steady-state problems where the ini-

tial condition u0 is generally strongly different from the final solution u∗ satisfying

L(u∗) = 0.

The discretization inspace of the pseudo time-step termin (5.20) is partially upwind

to stabilize the iteration process for high-speed flow calculations. To avoid stability

problems due to sudden changes at the boundaries, we use solution-correction de-

pendent underrelaxation of the boundary conditions that vanishes upon convergence.These two measures turn out to be very effective in practice.

The convergence behavior of the present method is as described in [24]: slow

convergence in the first or searching phase where the algorithm tries to get in the

neighborhood of u∗ (CF Lpseu = O(1)); fast convergence in the second or converging

phase where the algorithm feels the attraction of  u∗ (CF Lpseu 1). We have

observed that, depending on the difficulty of the system of equations to be solved,

the searching phase can be virtually absent or can take up to several hundreds of 

iterations. The converging phase invariably takes only about 20 to 30 iterations to

reduce the convergence error down to (maxi |d mi − d m−1i | < 10−11, maxi |qm

i /d mi −qm−1

i /d m−1i | < 10−13), the convergence criterion that we have used in all flow

calculations. This type of convergence behavior is for problems with steep-gradient

solutions where the underrelaxation of νart precludes quadratic convergence. Very

easy problems with smooth solutions that require virtually no artificial smoothing

converge faster.

An example of a system of equations that is difficult to solve is the calculation of a

complex steady-state flow through a complex geometry, modeled on a grid consisting

of 2000 grid points. The difficulty is entirely due to the fact that the initial condition(uniform discharge qW , constant water level h) is very different from the solution

that is sought. In contrast, a system is easy to solve whenever a reasonable initial

condition is available. This situation is encountered in unsteady calculations where

there is hardly any searching phase; the initial condition is the solution of the previous

physical time step and usually already very close to the solution of the next physical

time step. Flow calculations inside theadaptive grid algorithmarealso nearlyall easy.

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The compatible scheme that we apply consists of the superposition of a discretiza-

tion in space and a discretization in time (method of lines [MOL]). This permits

analysis of the error separately in computational space and in computational time.

5.5.1 Error Analysis in Space

The first step is to define a suitable smooth and infinitely differentiable approxima-

tionu of the piecewise linear numerical function approximation u that we have been

using. In principle that is easy since the space of infinitely differentiable functions is

dense in the space of piecewise polynomial functions. In practice it is slightly more

complicated because we need an algebraic description of that smooth function.

There are two reasonable ways to construct such a function: one is to connect grid-point values by a smooth function; the other is to connect cell-center values. Both

possibilities are illustrated in Figure 5.3 by, respectively, the functionu(γ  = 0) andu(γ  = 1). It can be seen that the closest smooth fit to u lies somewhere in between

these two functions.

FIGURE 5.3

Smooth approximations of a piecewise linear function.

This suggests consideration of the one-parameter family of functions obtained

by means of linear interpolation between u(γ  = 0) and u(γ  = 1), with γ  theinterpolation coefficient. It is not relevant to determine how these functions can be

constructed from the grid-point values of u. What matters is the inverse: u expressed

in terms of u and its derivatives at the grid points, since this is the information required

to construct the Taylor-series expansions. Skipping the details of the construction,

we will just present the result:

u(ξ) = u (ξ i−1 + βξ) = f −i (1 − β) − γ g−i (1 − β) + O

ξ 6

, (5.22)

or:

u(ξ) = u (ξ i−1 + βξ ) = f +i−1(β) − γ g+i−1(β) + O

ξ 6

, (5.23)

with ξ i−1 ≤ ξ  ≤ ξ i , i = 1, . . . , I  + 1, local coordinate β = (ξ  − ξ i−1)/ξ  ∈ [0, 1],

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and with:

f ±i (β) =ui + β

±ξ 

u i

i + ξ 2

2 u ii

i ± ξ 3

6 u ii i

i + ξ 4

24 u iv

i ± ξ 5

120u v

i ,

(5.24)

g±i (β) = ξ 2

8u ii

i ± βξ 3

8u ii i

i + (24β − 5)ξ 4

384u iv

i ± βξ 5

128u v

i . (5.25)

The superscripts in (5.24) and (5.25) indicate the order of differentiation with respect

to ξ  while subscript i indicates the grid point (e.g., u ii ii = ∂3u/∂ξ 3

ξ =ξ i

). Each

function in (5.22), (5.23), (5.24), and (5.25) is also a function of computational time

coordinate τ , but for clarity this has not been indicated.

For β = 1 expression (5.24) is the well-known Taylor-series expansion up toO(ξ 6) of grid-point value ui±1 in terms of derivatives at ξ  = ξ i when u passes

through the grid-point values (γ  = 0). The expressions (5.22) and (5.23) for γ  = 0

are obtained by combining this with u(ξ) = u(ξ i−1 + βξ) = (1 − β)ui−1 + βui

[cf. (5.13)].

The second term in the right-hand side of (5.22) and (5.23) can be viewed as a

γ -dependent correction that takes care of the shift required when the smooth approx-

imation is defined differently. From both (5.22) and (5.23) we obtain (expanding the

right-hand sides at ξ  = ξ i

−12):

u

ξ i− 1

2

=uξ 

i− 12

+ (1 − γ )

ξ 2

8u ii

i− 12

+ ξ 4

384u iv

i− 12

+ O

ξ 6

. (5.26)

This shows that γ  = 1 corresponds with a smooth function fit through the cell-

center values. In fact, the equation u(ξ i− 1

2) = u(ξ 

i− 12

) has been used to derive

the expression for g±i . Another equation that has been used in that derivation is

g+i−1(β) = g−

i (1 − β) + O(ξ 6). We also have f +i−1(β) = f −i (1 − β) + O(ξ 6).

The latter two equations must hold for the two expressions (5.22) and (5.23) to be

equivalent, and hence continuous at the grid points. This is trivial for γ  = 0 whenupasses through the grid-point values, but it is a condition to be imposed when deriving

the expression for g±i . It turns out that this, together with u(ξ 

i− 12

) = u(ξ i− 1

2) for

γ  = 1, fixes g±i completely.

The compatible discretization in space of the shallow-water equations has been

obtained upon the integration of the easy equations with added artificial smoothing

term over the finite volumes [ξ i− 1

2, ξ 

i+ 12], replacing each continuous function by its

piecewise linear approximation (Section 5.4.1). In other words, the discretized flow

equations L(u) = 0 can also be written as:

Li (u) = ξ 

i+ 12

ξ i− 1

2

L(u) dξ  = 0 , i = 1, . . . , I , (5.27)

where L stands for the easy shallow-water equations transformed to computational

space [Equations (5.11) and (5.12)], while this time u represents the collection of 

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finite volumes of different size in physical space. We see here why it is essential to

consider separate coordinate transformations for the different problems with which

we are dealing (see the beginning of Section 5.4). To make sure that the equivalent

equations will be equivalent with the discretized equations in physical space, weshould consider:

 xi+ 1

2

xi− 1

2

R(u)xξ 

d x = x

i+ 12

xi− 1

2

L(u)

xξ 

dx − x

i+ 12

xi− 1

2

L(u)xξ 

d x , i = 1, . . . , I . (5.30)

L(u)/xξ  = 0 and

L(

u)/

xξ  = 0 can be viewed as equations formulated in physical

space, cf. the equations in computational space (5.11) and (5.12) that have been

obtained upon multiplying the original equations (5.1) and (5.2) by xξ .

Using (5.26) to replace the integral boundaries xi− 1

2and x

i+ 12

of the second term in

the right-hand side, we obtain from (5.30) (recall thatxξ  andxi both denote ∂x/∂ξ ):

ξ 

R(u)|ξ i + O

ξ 2

=  x

i+ 12

xi− 1

2L(u)

xξ 

dx−  

xi+ 1

2+ 1−γ 

8 ξ 2

xii

i+ 12

+O

ξ 4

xi− 1

2+ 1−γ 

8 ξ 2xii

i− 12

+O(ξ 4) L(u)xξ 

d x= ξ 

i+ 12

ξ i− 1

2

L(u) dξ  − ξ 

i+ 12

ξ i− 1

2

L(u)dξ 

− 1 − γ 

8ξ 3

∂

∂ξ 

x iiL(u)xi

ξ i

+ O

ξ 5

, i = 1, . . . , I . (5.31)

The essential difference between (5.29) and (5.31) is the physical-to-computational-space correction in the right-hand side of (5.31). It vanishes for γ  = 1 when we have

[xi− 1

2, x

i+ 12] = [x

i− 12

,xi+ 1

2].

Since everything is now formulated in computational space again, thedetermination

of R is fairly straightforward. Term by term the finite volume integrals of L(u) and

of L(u) are expanded in a Taylor series at ξ  = ξ i , using (5.22) and (5.23) to replace

u by

u. For example, using the discretization in space (5.16), the evaluation of the

right-hand side of (5.31) for the time derivative ∂(xξ q)/∂τ  reads:

 ξ i+ 1

2

ξ i− 1

2

∂(xξ q)

∂τ dξ  −

 ξ i+ 1

2

ξ i− 1

2

∂(xξ q)

∂τ dξ  − 1 − γ 

8ξ 3

∂

∂ξ 

x iix i

∂(xξ q)

∂τ 

ξ i

= ξ 3∂

∂τ 

2 − 3γ 

24x iq ii + 1

24x iiq i + 1 − γ 

8x ii iq

ξ i

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− 1 − γ 

8ξ 3

∂

∂τ 

x iiq i +x ii iq− ∂

∂ξ 

xξ q ∂

∂τ 

x ii

x i

ξ i

+ O

ξ 5

=2

−3γ 

24 ξ 

3 ∂

∂τ  x i q ii

− xii

x i q iξ i

+ 1 − γ 

8ξ 3

∂

∂ξ 

q xiiτ  − x iix i

x iτ 

ξ i

+ O

ξ 5

. (5.32)

In the first term of the right-hand side of (5.32) one recognizes the second-order in-

terpolation error of q in physical space integrated over the finite volume [xi− 12

, xi+ 12]

and formulated in computational space:

 xi+ 12

xi− 1

2

(q −q ) d x =  ξ i+ 12

ξ i− 1

2

xξ q dξ  −  ξ i+ 12

ξ i− 1

2

xξ q dξ  − 1 − γ 

8ξ 3

∂x iiq∂ξ 

ξ i

= ξ 3

2 − 3γ 

24x iq ii + 1

24x iiq i + 1 − γ 

8x ii iq

ξ i

−1 − γ 

8ξ 3

x ii

q i +

x ii i

q

ξ i+ O(ξ 5)

= 2 − 3γ 

24ξ 3x i

i

q ii − x iix iq i

ξ i

+ O

ξ 5

. (5.33)

Using ∂2q/∂x2 = 1/xξ ∂(qξ /xξ )/∂ξ  = (q ii − x iiq i /x i )/(x i )2, this can also be

written as:

 xi+ 1

2

xi− 1

2

(q −q ) dx =2

−3γ 

24 x3

i

∂2q∂x2 ξ i+ O x

5

i ,

with xi = xi+ 12

− xi− 12.

The space discretization of the other terms in the flow equations is analyzed in

the same way. This shows for example that the second term in the right-hand side of 

(5.32) cancels against the part of the residual of the convection term ∂(q/d −xτ )q/∂ξ 

stemming from the interpolation error in physical space of xτ . Skipping the lengthy

derivation and reformulating the residual in terms of errors in the variables, we obtain

for the momentum equation:

∂

∂τ 

xξ 

q +

γ  − 1

24

Dξ (q)

+ ∂

∂ξ 

q + γ Dξ (q)d + γ Dξ (d)−xτ 

q + γ Dξ (q)

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 Err q = Err spaceq + Err time

q ≈ |Dξ (q)| + cγ |Dτ (q)| ≈ |Dξ (q)| + cγ |Dτ (q)| ,

(5.39)

with Dξ  and Dτ  as in (5.35) and (5.37). Since all functions are smooth (they shouldbe, by construction), the derivatives of d ,q, andx in Dξ  and Dτ  can be approximated

very well by the derivatives of d , q, and x using simple finite differences. We see

that, although the smooth fits of the piecewise polynomial numerical approximations

are the basis of the error analysis, these smooth functions are actually not required.

It suffices to know that they exist.

Scaling of the error expressions (5.38) and (5.39) with a constant coefficient is

irrelevant. However, the relative weighing between the interpolation error in space

and the one in time is important. It is determined by parameter cγ  that, in view of 

the coefficients of the errors in space in (5.34) and the coefficient of the error in time(5.37), should have a value of about (1/2)/(1/12) = 6.

The reliability of error approximations (5.38) and (5.39) depends to a large extent

on the smoothness of the bottom, cf. the discussion above definition (5.15) and below

equation (5.36). It may therefore be useful to consider instead of (5.38):

 Err h = Err spaceh + Err time

h ≈ |Dξ (h)| + cγ |Dτ (h)| ≈ |Dξ (h)| + cγ |Dτ (h)| .

(5.40)

For the determination of the artificial viscosity coefficient νart and the adaptationof the grid, the error in water depth d or water level h and in depth-integrated velocity

q should be combined in some way to a single error expression. A convenient option

is to base that combination on an energy measure like the energy head. Since the

energy head itself may be (nearly) constant [cf. expression (5.6)] we will consider the

sum of the error in the potential part and in the kinetic part of the flow energy. From

a physical point of view this seems to be a fairly meaningful choice. So for the grid

adaptation we consider the error expression:

 Err = Err space + Err time ≈ |Dξ (d)| + qDξ (q)

gd 2

− q2Dξ (d)

gd 3

+ cγ 

|Dτ (d)| +

qDτ (q)

gd 2

− q2Dτ (d)

gd 3

, (5.41)

which has been obtained by linearizing 12

q2/(gd 2

) − 12

q 2/(g

d 2), the interpolation

error in the kinetic part 12

u2/g of the energy head, with respect to Dξ (q), Dξ (d),

Dτ (q), and Dτ (d).The dimension of  Err is [m]. Therefore, expression (5.41) cannot be used for Err ν ,

the error in the right-hand side of artificial viscosity equation (5.14). Instead we use:

 Err ν = ξ xξ 

1

2

 g

d |Dξ (h)| +

 1

2

Dξ (q)

d − qDξ (d)

d 2

, (5.42)

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obtained by linearizing the interpolation error in the square root of the potential part

and the kinetic part of the energy head, multiplied by√ 

g and by grid size ξ xξ  . Only

the interpolation error in space is considered; it is not necessary to include the error in

time in the error feedback mechanism because the dissipative backward Euler schemethat we use provides enough smoothing in time. It is easily verified that (5.42) is as

described below Equation (5.14) and satisfies the artificial dissipation requirements

mentioned in Section 5.2.

Notice that the interpolation error in the water depth is used in (5.41), while in

(5.42) we look at the interpolation error in the water level. Ideally, these two should

be nearly equivalent; at present they are not because of the absence of geometry

smoothing. Since a smooth water surface implies a smooth hydrostatic pressure term

in (5.2) and hence a smooth flow, the use of Dξ (h) is preferred in (5.42). This avoids

unnecessarily large values of νart occurring in regions with large variations in waterdepth but with small flow velocities. However, numerical experiments have shown

that it is best to use Dξ (d) and Dτ (d) in (5.41), probably because in this way the

algorithm “feels” to some extent the nonsmoothness of the bottom. All this is rather

a matter of compromising and certainly not ideal, which is confirmed by the results

that we will present.

5.6 Error-Minimizing Grid Adaptation

In the previous section we derived expression (5.41): a physically meaningful

approximation of the numerical modeling error.  Err is a function of the numerical

solution that is sought, as well as of the piecewise linear coordinate transformation

x(ξ,τ) defined by the grid point coordinates xni [cf. (5.10)]. The grid points form a

set of degrees of freedom that can be chosen in any convenient way. Here, we choose

to determine them by considering the optimization problem:

solve: minx(ξ,τ)

Err 1 . (5.43)

Numerical modeling error  Err is measured in L1-norm for reasons explained in

Section 5.1. Putting (5.41) and (5.43) together, we obtain:

solve: minx(ξ,τ)

 t N 

t 0

 xright

xleft  Err space + Err time

dx dt 

= solve: minx(ξ,τ)

 t N 

t 0

 ξ I + 12

ξ 12

xξ  Err space + Err time dξ dt ,

(5.44)

with xleft and xright the coordinates of the left and right boundary in physical space, and

ξ 12

and ξ I + 1

2the coordinates of these boundaries in computational space [cf. Figure 5.2

and (5.10)].

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get the grid fully converged under all circumstances (see the next section). With each

outer grid iteration we solve the flow equations given the grid defined by x(ξ,τ), and

solve the grid equations to obtain x(ξ , τ ) given the flow solution. Both are iterative

solution procedures; the former has been described in Section 5.4.2, while the latteris the inner grid iteration loop described below.

For clarity, we will explain the grid optimization procedure for a single unknown

a. Approximating the integral in time by means of a one-point quadrature rule, the

optimization problem to be solved per time step is:

minxn(ξ )

t 

 ξ I + 1

2

ξ 12

xn− 1

2

ξ 

ξ 2|aξ ξ  − xξ ξ 

xξ aξ  | + cγ τ |aτ  |

n− 12

dξ  , (5.46)

where we have used an expression like (5.40) for a, substituting (5.35) and (5.37).

For the optimization of the grid in shallow-water applications, the more complicated

error expression (5.41) is used, but the principle remains the same.

Optimization problem (5.46) has been formulated in the computational space

(ξ , τ ) corresponding with optimal coordinate transformation x(ξ , τ ). To be able

to solve it, we transform it to the current computational space ( ξ , τ ), introducing

ξ ( ξ , τ ) and τ  = τ , the transformation from current to optimal computational space.

It is most convenient to consider ξ ( ξ , τ ) and not x(ξ , τ ) as the unknown function

to be determined, since ξ ( ξ , τ ) is defined (and hence can be approximated) on thecurrent computational grid. The coordinates of the optimal computational space cor-

responding with the grid points in the current computational space will be denoted

by ξ ,ni = ξ (ξ i , τ n). Notice that ξ 

,ni − ξ 

,ni−1 is not a grid size and hence in general

not equal to ξ , the size of the uniform grid in the optimal computational space.

At this point we can take the size of the grid in computational space (any compu-

tational space) equal to any convenient value. We will use ξ  = ξ  = 1 [recall

that we have τ  = τ  = t  because of (5.9)]. The position of the boundaries of 

a computational domain are fixed, ξ 12 =

ξ 12 =

1

2

and ξ I +

1

2 =ξ 

I +1

2 =I 

+1

2

, and so

optimization problem (5.46) formulated in ( ξ , τ ) becomes:

solve: minξ ,n(ξ )

t 

 ξ I + 1

2

ξ 12

xn− 1

2ξ 

|Dξ (a)|

(ξ ξ )2

+ cγ τ |aτ  −ξ τ 

ξ ξ 

aξ |n− 1

2

dξ . (5.47)

In order to be able to solve this problem we assume that a is a function of the

physical coordinates only, and independent of the grid size. This is obviously not true

in general, especially in regions of steep gradients, but a reasonable approximation forsmall grid perturbations (in particular perturbations around the optimal grid) because

of the smoothness of the solution. Since we have fixed momentarily coordinate

transformation x(ξ,τ), this implies that x and a in (5.47) are both considered to be

independent of ξ .Under this assumption it is straightforward to solve (5.47). The integral is ap-

proximated by a sum over the grid cells using straightforward discretizations, and

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differentiated with respect to the ξ ,ni , i = 1, . . . , I  . Equating the result to zero, the

system of equations is obtained that defines the optimal grid:

8xn−12

i+ 12

(ξ ,ni+1 − ξ 

,ni + 1)3

eξ,n− 1

2

i+ 12

−8xn−

12

i− 12

(ξ ,ni − ξ 

,ni−1 + 1)3

eξ,n− 1

2

i− 12

= cγ τ 

2xn− 1

2

i+ 12

(ξ ,ni+1 − i)

(ξ ,ni+1 − ξ 

,ni + 1)2

eτ,n− 1

2

i+ 12

−2x

n− 12

i− 12

(ξ ,ni−1 − i)

(ξ ,ni − ξ 

,ni−1 + 1)2

eτ,n− 1

2

i− 12

,

i = 1, . . . , I , (5.48)

with:

eξ  = smoothed |aξ ξ  − aξ xξ ξ /xξ | ,

eτ  = smoothed aξ  tanh

τ(aτ  − aξ ξ τ /ξ ξ )

cτ |a|

.

(5.49)

The latter is an approximation of eτ 

=sign(aτ 

−aξ ξ 

τ 

/ξ ξ 

) that, not surprisingly,

turns out to lead to serious stability problems. The argument of the tanh function is

scaled with |a| to make it non-dimensional. For the shallow-water equations we scale

with the energy-head-like expression d + 12

q2/(gd 2

) and use cτ  = 10 for the scaling

parameter. The smoothing of eξ  and eτ  is the same as the one applied for νart, i.e.,

like Equation (5.14) whose discretization is given in (5.18), using the same value of 

constant smoothing coefficient α.

Equation (5.48) with boundary conditions 12

(ξ ,n0 +ξ 

,n1 ) = 1

2and 1

2(ξ 

,nI  +ξ 

,nI +1) =

I 

+1

2

is solved iteratively by means of a Newton-type method, evaluating eτ  explicitly

using a very strong underrelaxation. The diagonal of the implicit part of the linearized

system of equations per iteration is increased for additional stability. The approach

is similar to that used for the flow equations (see Section 5.4.2). Although we obtain

convergence down to 10−10, the convergence is usually slow and certainly needs to

be improved.

Once Equation (5.48) has been solved, we use the grid point values ξ ,ni to define

the piecewise linear function ξ ,n

(ξ ). The coordinates of the next approximation of 

the optimal grid are determined by x,ni

=xn(ξ ), with ξ  the solution of ξ 

,n(ξ )

=i, i = 1, . . . , I  . In practice, we do not use xn(ξ ) here but a smooth approximationbased on a monotonicity-preserving cubic Hermite interpolation [6]. This leads to

smoother grid updates in the outer grid iteration loop without changing the final, fully

converged grid. The coordinates of the virtual grid points are obtained from the grid

boundary conditions 12

(x,n0 + x

,n1 ) = xleft and 1

2(x

,nI  + x

,nI +1) = xright.

Once the outer grid iteration loop has converged we have ξ ,ni → ξ i = i in which

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than directly by means of the artificial viscosity mechanism, but this is not true. This

is because of the way the numerical solution method attempts to approximate the

infinitely steep gradient at the top two edges (see the exact water level in Figure 5.4).

The artificial viscosity takes care of these difficult details, but without additional ge-ometry smoothing this mechanism is unable to provide enough smoothing, especially

on a very fine grid when the details are felt strongly. The net result is that on very

fine grids small wiggles tend to become important, thereby violating the assumption

underlying the whole method that higher-order errors should be negligible. See also

the adaptive grid results below.

The effect of the artificial viscosity on the solution is evidenced by the differ-

ence between the exact water level and its numerical approximation (see Figure 5.4).

In practical applications, an exact solution would not be available. A comparison

between the artificial viscosity and other modeling terms is however always possi-ble. This information becomes especially useful once bottom friction and a turbulent

viscosity model will be present, since it indicates immediately if deviations from

measurement data are to be attributed to physical or numerical modeling errors. See

also Section 5.2.

The energy head in Figure 5.4 downstream and upstream of the hydraulic jump

is nearly constant, in agreement with the exact solution (see Section 5.3). Also the

energy undershoot of a viscous hydraulic jump is predicted by the model. Not shown

is the mass conservation “error”. Because continuity equation (5.1) is discretized overthe finite volumes [ξ i− 

12

, ξ i+ 

12], the mass flow at the cell centers is constant within

O(10−12) which is the remaining convergence error. Perfect mass conservation at

the discrete level does not prevent, however, a difference of 0.0352% between the

value of q at the even-numbered grid points and the value of q at the odd-numbered

grid points. This small mass flow wiggle is mainly due to the nonsmoothness of the

geometry.

Solving the steady-state grid adaptation equation (5.52) and the flow equations

alternately in 50 outer grid iterations gives the fully converged adaptive grid result

shown in Figure 5.5. The outer grid iteration loop starts with a maximum relative grid

correction maxi |ξ i − ξ i | = maxi |ξ i − i| of 23.5 in the first iteration, and reduces it to

4.72 × 10−6 in the last iteration. With such small grid corrections, there is of course

no longer a gain in numerical accuracy. Comparing the exact solution and numerical

solution in L1 norm, it appears that the grid can be considered fully converged when

maxi |ξ i − i| < 0.1. This grid convergence criterion is reached after 13 outer grid

iterations. The numerical solution error is then within 1% of its value on the fully

converged grid.

A comparison between Figure 5.4 and Figure 5.5 clearly shows the significantaccuracy improvement. The energy head upstream and downstream of the hydraulic

 jump is virtually constant, there is hardly any difference visible between the exact

solution of the water level and its numerical approximation, and the level of the

artificial viscosity is very low except at the locations where the solution is not smooth.

However, the mass conservation error is 0.0156% which is barely a factor 2 better

than on the uniform grid.

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FIGURE 5.5

Shallow-water solution on adapted grid, 100 finite volumes.

A detail of the adaptive grid solution is shown in Figure 5.6 together with theuniform grid solution. It can be observed that the structure of the numerical approx-

imation of the hydraulic jump on the adapted grid is identical to the one shown in

Figure 5.4. The jump is again spread over about 5 to 6 grid cells while the levels of 

the undershoot of the energy head are the same, indicating that the artificial viscosity

mechanism has been dimensioned correctly.

Parameter value α = 3 has also been used for the smoothing of the error Err space

[see (5.41)] that has been minimized. With this value, the grid stretching is limited to

78.9% or|ln(x

i+ 

1

2

/xi− 

1

2

)| ≤

0.582. This result is obtained from discretization

(5.18) of smoothing equation (5.14) that limits the rate of decay of the smoothed

interpolation error to a factor ( 34+2α+2

 18

+ α)/(2α− 14

) per grid cell. Substituting

α = 3 in this factor gives 1.789 (ln 1.789 = 0.582), which is also the maximum grid

stretching since the grid size is inversely proportional to the smoothed error [the

equidistribution principle, Equation (5.52)].

The grid size and grid stretching as a function of computational space coordinate

ξ  are shown in Figure 5.7, with the grid size non-dimensionalized by the size of the

uniform grid xunif 

=5 m. One recognizes the moderate grid refinement near the

four edges of the bar (a stronger refinement near the top two edges) and the large

refinement near the hydraulic jump.

The maximum stretching is indeed reached at a number of places. However, the

maximum stretching is not reached at the boundaries where we would have expected

it. The solution near the boundaries is uniform, the interpolation error becomes zero,

and so the decay of the smoothed interpolation error and hence the grid stretching

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FIGURE 5.7

Computed optimal coordinate transformation for steady-state shallow-water

calculation, 100 finite volumes.

Table 5.1 Convergence Behavior of Adaptive Grid

Shallow-Water Calculations

# Vols # Iters xmin xmax h − h1 Order CPU (sec.)

25 8 5.79E0 109.4 2.59E-1 0.38

35 9 2.22E0 100.9 1.81E-1 1.06 0.44

50 11 7.29E-1 92.7 9.61E-2 1.78 0.77

70 11 2.32E-1 78.4 4.07E-2 2.55 1.10

100 13 5.96E-2 74.4 1.34E-2 3.31 1.48

140 25 1.81E-2 29.7 5.15E-3 2.68 3.90

200 30 6.71E-3 11.3 2.27E-3 2.30 7.09

280 50 3.44E-3 7.3 1.01E-3 2.40 24.22

400 50 2.49E-3 3.8 5.43E-4 1.74 40.15

were rounded over a length of 10 m. This is, however, only a partial solution to

the problem since pointwise geometry approximation (5.15) is not able to “see” the

curved shape of the smoothed edges in between the grid points. As a consequence,

each time the grid isadapted and the points moveto a differentposition, the flow solver

sees a slightly different geometry and hence calculates a slightly different solution.

This is obviously not favorable to the grid convergence process and explains why the

results were only marginally better.

A series of tests with α = 6, in an attempt to compensate for the non-smooth

geometry by increasing the artificial viscosity smoothing, was more successful. This

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time we also obtained grid convergence when 280 volumes were used, and lower

errors for 200 finite volumes and more, despite the fact that the grid was less refined

near the hydraulic jump (the location of minimum grid size xmin). From this we

conclude that the adaptive grid modeling of the discontinuous jump is not posing aproblem. Since the hydraulic jump is at a position where the geometry is smooth, this

is consistent with our interpretation. The solution error was, however, larger when

less than 200 volumes were used, because with α = 6 the grid stretching is limited

to 51%.

The last column in Table 5.1 gives the CPU time of the calculations that were

all executed on a 400-MHz Pentium notebook computer. To assess the efficiency

of the adaptive grid method they should be compared with the CPU time of the

same calculation on a uniform grid. Steady-state shallow-water calculations on a

uniform grid are, however, very fast, mainly because of the effort that we have spent in

optimizing the iterative flow solver (see Section 5.4.2). For example, a fully converged

steady-state calculation for the same problem on a uniform grid of 400 finite volumes

takes only 0.82 CPU sec. The L1 error in the water level, h − h1, turns out to be

2.06E-2 for this calculation. This result is more accurate and obtained faster than

the result of the adaptive grid calculation with 70 finite volumes (see Table 5.1). On

the other hand, the adaptive grid convergence criterion that we applied was rather

severe. There is also still room for improvement of the adaptive grid solver (see, e.g.,

Section 5.4.2). We especially expect a significant gain in accuracy and efficiencyonce geometry smoothing is implemented.

5.7.2 Unsteady Application

We present an unsteady adaptive grid application that, strictly speaking, is not even

a genuine unsteady application. It is a fairly simple and yet extremely complicated

test for moving adaptive grid methods, and clearly shows both the advantages and

shortcomings of our moving adaptive grid approach.The test is simple: starting from the steady-state solution shown in Figure 5.4,

calculate the steady-state solution on the adapted grid of  Figure 5.5 by solving the

flow equations both in time and space using a time step of 1 sec. Although a steady-

state problem in physical space, it is not  a steady-state problem in computational

space where we solve (5.11) and (5.12).

We will first solve this problem in the standard way, applying the equidistribution

principle in space and ignoring the effect that this has on the error in computational

time. This is realized by solving per outer grid iteration Equation (5.48) with cγ  = 0.As can be seen in Figure 5.8, the effect is dramatic. Virtually all grid points are drawn

toward the hydraulic jump region in the very first time step. As a consequence, many

grid points move to a position with a totally different water depth and velocity. This

leads to very large variations in computational time and, since time derivatives and

space derivatives are coupled, also to very large perturbations of the solution in space.

This applies in particular to grid points at and in the vicinity of the bar.

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grid points at the base of the bar which is due to the fact that the grid points are again

drawn toward the hydraulic jump. The difference with the previous result without

time error compensation is that this time the error minimization in time prevents the

grid points from crossing the bar.The overshoot in the water level at the base isprobablydue to the non-smoothnessof 

the geometry. The results obtained in subsequent time steps show that this overshoot

initiates a small wave moving essentially upstream and decaying rapidly. However,

grid points start following that waveand are then not available to improve the accuracy

elsewhere. We have observed that the solution error h − h1 did decrease from the

very first time step, although very slowly (less than a factor 2 in 20 time steps) due to

the perturbations introduced in the first time step.

Nofull convergencehas beenobtainedfor the adapted grids of thissection, although

we did not spend much effort in finding suitable values for the different convergenceparameters. We found that rather useless because of the fundamental shortcoming

already mentioned in Section 5.6: the lack of adaptive flexibility when the grid is

continuous in time. The results presented here reveal that this is a major drawback in

shallow-water applications where grid points may have to move continuously across

non-uniform parts of a channel geometry in order to get a high resolution in certain

dynamically changing regions. This will always lead to relatively large errors in time

and will always require some compensation mechanism. In our method the com-

pensation is included automatically, based on the error in time and only active when

necessary. Nevertheless, this compensation (and any such compensation mechanism)

will inevitably limit the grid speed and hence the potential gain in accuracy. There

may then be no advantage in using a moving adaptive grid technique.

The solution to this problem has already been suggested in Section 5.6: consider a

grid that is discontinuous in time and minimize the error per time step (5.45) also with

respect to xn−1(ξ ). The derivation of the optimal grid equations is straightforward

and leads to an interesting result that can be summarized as follows. Average grid size

xn− 1

2

i

−x

n− 12

i

−1 will be optimized for minimal error in space, while grid displacement

xni − xn−1

i will be optimized for minimal error in time. This virtually independentminimization of the numerical modeling error in space and time indicates that a

discontinuously moving adaptive grid method may prove to be very efficient.

5.8 Conclusions

We have presented the development of a moving adaptive grid method that aimsat minimizing the numerical modeling error (the part of the numerical solution error

generated locally as a result of solving the model equations numerically), rather

than the numerical solution error. Besides being virtually impossible to realize for

problems of practical interest, the usefulness of the latter is limited because of the

presence of physical modeling errors and data errors whose effect should also be

taken into account.

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The idea behind the present approach is first to ensure that physics-based artifi-

cial smoothing terms form the dominant numerical modeling error to allow a direct

comparison with physical modeling terms; and second to minimize the effect of the

artificial smoothing terms by means of grid adaptation. We have shown that this isfeasible in 1-D, although the presented numerical algorithm is still incomplete. One

element that needs to be added is geometry smoothing, the importance of which is

illustrated by the results.

Smoothness is the key element of the proposed method; it ensures that the effect

of discretization errors on the numerical solution is small, which is essential for a

meaningful error analysis. The use of a compatible scheme is required to be able to

analyze that effect, and to obtain a useful approximation of the numerical modeling

error in space and in time as a function of the local grid parameters. The combination

of artificial smoothing and a compatible discretization has enabled us to developan error-minimizing moving adaptive grid algorithm, minimizing the effect of both

discretization errors and smoothing errors.

The results clearly show the importance of taking the error in time into account

when adapting the grid in space. However, grid adaptation can be very inefficient if 

the grid is forced to move continuously. This applies in particular to the unsteady

shallow-water applications with non-uniform bottom in which we are interested. An

extension that needs tobeconsidered is thereforethedevelopment ofa discontinuously

moving adaptive grid method.

The complexity of the developed method is large. On the other hand, a high gain in

accuracy is possible since the method is capable of using grid points very efficiently.

For unsteady calculations this will only be true after the method has been extended

with a grid that can move discontinuously in time.

There are indications that the 1-D error analysis of the compatible scheme that

we have presented can be extended to several space dimensions, showing the corre-

spondence between the multi-D numerical modeling error and multi-D interpolation

errors. Multi-D interpolation errors depend however in a complicated way on the grid

size, grid stretching, grid curvature, and grid skewness. Minimizing these errors bymeans of grid adaptation will be very difficult to realize. On the other hand, elements

of such a technique may possibly be combined with a more heuristic adaptive grid

approach to arrive at better monitor functions and hence more efficient grid adaptation

procedures for multi-D applications of practical interest.

References

[1] M. Arora and P.L. Roe, On postshock oscillations due to shock capturing

schemes in unsteady flows, J. Comput. Phys., 130, (1997), 25–40.

[2] M. Borsboom, Development of an error-minimizing adaptive grid method,

 Appl. Num. Math., 26, (1998), 13–21.

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[3] M.H. Chaudhry, Open-Channel Flow, Prentice-Hall, 1993.

[4] V.T. Chow, Open-Channel Hydraulics, McGraw-Hill, 1973.

[5] E.A. Dorfi and L.O’C. Drury, Simple adaptive grids for 1-D initial value prob-lems, J. Comput. Phys., 69, (1987), 175–195.

[6] R.L. Dougherty, A. Edelman, and J.M. Hyman, Nonnegativity-, monotonicity-,

or convexity-preserving cubic and quintic Hermite interpolation, Math. Comp.,

52, (1989), 471–494.

[7] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive

methods for differential equations, Acta Numerica, 4, (1995), 105–158.

[8] K. Eriksson and C. Johnson, Adaptive streamline diffusion finite element meth-ods for stationary convection-diffusion problems, Math. Comp., 60, (1993),

167–188.

[9] R.M. Furzeland, J.G. Verwer, and P.A. Zegeling, A numerical study of three

moving-grid methods for one-dimensional partial differential equations which

are based on the method of lines, J. Comput. Phys., 89, (1990), 349–388.

[10] P. Garcia-Navarro, F. Alcrudo, andJ.M. Saviron,1-Dopen-channelflow simula-

tion using TVD-McCormack scheme, J. Hydr. Engrg., 118, (1992), 1359–1372.

[11] T. Geßner, D. Kröner, B. Schupp, and M. Wierse, Finite volume meth-

ods for conservation laws and convection-dominated diffusion equations, in:

F. Benkhaldoun and R. Vilsmeier, eds., Finite Volumes for Complex Applica-

tions — Problems and Perspectives, Hermes, Paris, 1996, 61–76.

[12] D.F. Hawken, J.J. Gottlieb, and J.S. Hansen, Review of some adaptive node-

movement techniques in finite-element and finite-difference solutions of partial

differential equations, J. Comput. Phys., 95, (1991), 254–302.

[13] C. Hirsch, Numerical Computation of Internal and External Flows, Volume 2,

Wiley, 1990.

[14] P. Houston, J.A. Mackenzie, E. Süli, and G. Warnecke, A posteriori error anal-

ysis for numerical approximations of Friedrichs systems, Numer. Math., 82,

(1999), 433–470.

[15] P. Houston, R. Rannacher, and E. Süli, A posteriori error analysis for stabilised

finite element approximations of transport problems, Tech. Rep. NA-99/04, Ox-

ford University Computing Laboratory, 1999.[16] W. Huang and R.D. Russell, Analysis of moving mesh partial differential equa-

tions with spatial smoothing, SIAM J. Numer. Anal., 34, (1997), 1106–1126.

[17] C. Johnson, R. Rannacher, and M. Boman, Numerics and hydrodynamic sta-

bility: toward error control in computational fluid dynamics, SIAM J. Numer.

 Anal., 32, (1995), 1058–1079.

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[18] A.A. Khan and P.M. Steffler, Physically based hydraulic jump model for depth-

averaged computations, J. Hydr. Engrg., 122, (1996), 540–548.

[19] R.J. LeVeque and H.C. Yee, A study of numerical methods for hyperbolic

conservation laws with stiff source terms, J. Comput. Phys., 86, (1990), 187–210.

[20] J.A. Mackenzie, The efficient generation of simple two-dimensional adaptive

grids, SIAM J. Sci. Comput., 19, (1998), 1340–1365.

[21] E.A. Meselhe, F. Sotiropoulos, and F.M. Holly Jr., Numerical simulation of 

transcritical flow in open channels, J. Hydr. Engrg., 123, (1997), 774–783.

[22] Y. Qiu and D.M. Sloan, Numerical solution of Fisher’s equation using a moving

mesh method, J. Comput. Phys., 146, (1998), 726–746.

[23] T. Sonar and E. Süli, A dual graph-norm refinement indicator for finite volume

approximations of the Euler equations, Numer. Math., 78, (1998), 619–658.

[24] B. van Leer and W.A. Mulder, Relaxation methods for hyperbolic equations,

in: F. Angrand, A. Dervieux, J.A. Desideri, and R. Glowinsky, eds., Numerical

 Methods for the Euler Equations of Fluid Dynamics, SIAM, 1985, 312–333.

[25] J.G. Verwer, J.G. Blom, R.M. Furzeland, and P.A. Zegeling, A moving grid

method for one-dimensional PDEs based on the method of lines, in: J.E. Fla-herty, P.J. Paslow, M.S. Shephard, and J.D. Vasilakis, eds., Adaptive Methods

 for Partial Differential Equations, SIAM, 1989, 160–175.

[26] J. von Neumann and R.D. Richtmyer, A method for the numerical calculation

of hydrodynamic shocks, J. Applied Physics, 21, (1950), 232–237.

[27] G.P. Warren, W.K.Anderson, J.L. Thomas, andS.L. Krist, Grid convergence for

adaptive methods, in: M.J. Baines and K.W. Morton, eds., Numerical Methods

 for Fluid Dynamics 4, Oxford University Press, 1993, 317–328.

[28] F.M. White, Viscous Fluid Flow, McGraw-Hill, 1974.

[29] Y. Xiang, N.R. Thomson, and J.F. Sykes, Fitting a groundwater contaminant

transport model by L1 and L2 parameterestimators, Adv. Water Res., 15, (1992),

303–310.

[30] P.A. Zegeling, Moving-grid methods for time-dependent partial differential

equations, CWI-tract No.94, Centre for Math. and Comp. Science, Amsterdam,

1993.

[31] P. Zegeling, M. Borsboom, and J. van Kester, Adaptive moving grid solu-tions of a shallow-water transport model with steep vertical gradients, in: V.N.

Burganos, ed., Proc. 12th Int. Conf. on Comput. Methods in Water Resources,

Vol. 2, Computational Mechanics Publications, 1998, 427–434.

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also been established in other river basins. In this chapter, the model of the River

Elbe is described. It is based on a dead-zone approach, consisting of a parabolic

convection-diffusionequation coupled by an additional linearequation, which models

the exchange of mass concentration between the main stream and the dead zones [13].In Figure 6.2 the influence of the different processes transport, diffusion, linear decay,

and dead zones is sketched. The related equations are presented in the next section.

FIGURE 6.2

Influence of different processes on the concentration (c(x,t 0) describes the

initial condition and c(x,t 1) is the resulting concentration at time t 1 > t 0),

(a) convection-diffusion model, (b) dead-zone model.

The final example requires a two space-dimensional modeling. The task is to

compute the area, which will be flooded in consequence of a specific flow discharge.

Usually, the flow discharge is related to a return period, e.g., a 100-year flood. The

flooded area is then shown on a map, in order to derive the flood risk for a building or

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a street. In Figure 6.3 such an example is shown for a small reach of the lower River

Saar, a tributary of the Moselle.

FIGURE 6.3

Flooded area in the lower River Saar.

6.2 Modeling Flow and Transport in Rivers

River flow modeling in two space dimensions is usually based on the 2D shallow-

water equations (2D SWEs). They can be derived from the Navier–Stokes equations

by assuming a hydrostatic pressure law. The 2D SWEs in conservative variables

read [39]

qt  + e(q)x + g(q)y = s(q) . (6.1)

q = h

uh

vh

is the vector of states with water depth h(t,x,y) and depth averaged

velocities u(t,x,y),v(t,x,y) in x (resp. y) direction.

The flux vectors are given bye(q)= uh

u2h + 12

gh2

uvh

and g(q)=

vh

uvh

v2h + 12

gh2

.

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The source term s(q) =

0

gh (S 0x − S f x )

gh (S 0y − S fy )

accounts for bottom friction and bot-

tom slope.

The expression S fx (resp. S fy) is called friction slope. An empirical formula (by

Manning–Strickler) reads

S f x = 1

K2S h

43

· u · 

u2 + v2, S fy = 1

K2S h

43

· v · 

u2 + v2, KS  ∈ R+ .

The constant KS  depends on the soil condition of the riverbed and is called roughness

coefficient. It is assumed that the friction slope implicitly accounts for the main

effects of turbulence as well.

The bottom slope is given by S 0i = −∂i b(i = x , y ) with the bottom elevationb(x,y). In Figure 6.4 the basic notations are shown.

FIGURE 6.4

Basic notations.

The Saint–Venant’s equations are the shallow-water equations in one space dimen-

sion x directed along the river course. The water surface elevation in this case is given

by z(t,x) = b(x) + h(t,x), where b(x) is the bottom level, S 0 := −bx is the bottom

slope, and h(t,x) is the water depth.

For arbitrary cross-sections A(t, x) the equations read [34]:

Conservation of mass: At  + (uA)x = 0 (6.2)Conservation of momentum: (uA)t  +

u2A

x

= −gA

zx + S f 

.(6.3)

The friction slope is given by

S f  = 1

K2S R

43

· |u| · u (6.4)

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where KS  is Strickler’s roughness coefficient and the hydraulic radius R is the ratio

of the cross-section area and the wetted perimeter. R can be well approximated by

R ≈ h for wide river cross-sections.

Supplied with data of the cross-section profiles in the form of paired values (z, A),the water-depth h(t,x) can be determined at each location x as a function of the

cross-sectional area A(t,x) in some way: h(t,x) = H(x,A(t,x)), e.g., by linear

interpolation. Then

zx = bx + ∂H 

∂x+ ∂H 

∂AAx

and Equation (6.3) may be written as

(uA)t 

+ u2Ax +

gA∂H 

∂A

Ax

= −gA

∂H 

∂x +gA S 0

−S f  . (6.5)

Equations (6.2) and (6.5) with the variable Q=uA lead to the hyperbolic systemA

Q

t 

+

0 1

gA ∂H ∂A

− QA

22 Q

A

A

Q

x

=

0

gA(S 0 − S f  − ∂H ∂x

)

(6.6)

whose eigenvalues are λ1,2 = QA

±c. (c = 

gA ∂H ∂A

is the wave speed.)

If both eigenvalues have the same sign, the flow is called supercritical, otherwise

it is subcritical (cf. supersonic, subsonic for the Euler equations of gas flow).A rectangular prismatic channel of constant width B with

A(t,x) = B · h(t,x) ⇒ ∂H 

∂A= 1

B= const.

defines a special case of the Saint–Venant equations:

∂t 

h

uh

+ ∂x

uh

u2h + 12

gh2

=

0

gh(S 0 − S f )

. (6.7)

The transport of soluble substances in natural rivers is usually described in 1Dthrough a simple convection-diffusion-reaction approach:

ct  = −ucx + DLcxx − KAc , (6.8)

withconcentration c(x,t), flow velocityu, dispersion coefficient DL, and linear decay

rate KA.

A more realistic model is obtained if dead zones are taken into account where very

low flow velocities occur. Because of concentration exchange with these dead zones,

the concentration curves do not remain symmetric in nature. These quasi-2D effectscan be modeled by a linear exchange term in the equation for the concentration c and

by adding a second equation for the concentration in the dead zone s [13, 25]:

ct  = −ucx + DLcxx − A0

AK(c − s) − KAc , (6.9)

st  = K(c − s) − KAs (6.10)

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with concentration c(x,t) in the main river, concentration s(x,t) in the dead zone,

area ratio dead zone/main riverA0A

, and exchange rate K .

All equations have to be completed by appropriate boundary and initial conditions.

6.3 Method of Lines Approach

6.3.1 Network Approach

In technical simulation of time-dependent processes, today’s industrial software

is often based on a network approach. This approach is embedded in the technicalcomputer-aided design (TCAD) environment and allows automatic generation of the

mathematical models [12]. In river simulation problems, a similar approach can

be applied. The network elements are 1D or 2D models for certain river reaches,

coupling elements like weirs, or junctions with tributaries and boundary elements like

gauging stations [24]. The single river reaches are modeled by the partial differential

equations presented in the upper section. All other elements are represented by

algebraic equations. By using the method of lines approach (MOL) for the partial

differential equations, a large system of differential algebraic equations (DAEs) is

generated [27, 25]. For its time integration, standard DAE-software can be applied.This process is demonstrated via the following one-dimensional example. All

1D equations may be summarized in

qt  = f  (t , x , q , qx ) , (6.11)

defined on the time interval t 0 ≤ t  ≤ t 1 and space interval α ≤ x ≤ β. q =(q1, . . . , qn), qi := qi (x,t), i = 1, . . . , n aretheunknownswith qt  = (

∂q1

∂t , . . . ,

∂qn

∂t )t ,

qx = (∂q1

∂x, . . . ,

∂qn

∂x) and f  = (f 1, . . . , f  n)t .

The following investigations are based on a semidiscretization in space by finite

differences but they are transferable to finite volume discretizations.

Let q i (t) = q(Xi , t) be defined on the equidistant space-mesh Xi = α + ix, i =0, . . . , N  with meshsize x = α−β

N .

Approximation of 

qx (Xi , t ) = qi+1(t) − q i−1(t )

2x

and substitution into (6.11) yields a linear implicit ODE-system of dimension n(N +1):

Ady

dt = f ( t , X0, . . . , XN , y) , with (6.12)

y =

q01 , . . . , q0

n , . . . , q N 1 , . . . , q N 

n

, A = diag

A0, I , . . . , I , AN 

.

A0 and AN  depend on the prescribed boundary conditions. In case of the maximum

number of 2n boundary conditions, one has A0 = AN  = 0. If no boundary condition

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has to be satisfied, then A0 = AN  = I , I  is the identity matrix. In this case

central finite differences must be replaced by forward differences in α and backward

differences in β. The function f  depends on f , the space discretization, and the

boundary conditions. The initial conditions at time t 0 must be consistent with theboundary conditions in α and β [25].

Theconvergence of thesemidiscretized system(6.12) to theexact solutionof (6.11)

for N  → ∞ is presumed.

6.3.2 Space Discretization

To prevent the numerical solution from spurious oscillations, the space discre-

tization must be adapted to the hyberbolic character of the flow equations. Moreover,

in a 2D model, complex river geometries must be handled properly. This is possiblethrough conservative finite volume (FV) schemes.

We start with the initial boundary value problem

qt  + f(q)x = s(q) , (6.13)

q(0, x) = q0(x) , (6.14)

q(t,α) = qα(t) , q(t, β) = qβ (t) , (6.15)

where the space-interval [α, β] is partitioned in N  cells I 1, . . . , I  N  through a given

set of N +1 mesh-points by α = x 12 < · · · < xN + 12 = β.Integration of (6.13) over thecontrol volume I j  = [x

j − 12

, xj + 1

2] withlength xj  =

xj + 1

2− x

j − 12

yields xj + 1

2

xj − 1

2

∂t q(t,x)dx = −

f 

q

t, xj + 1

2

− f 

q

t, xj − 1

2

+ x

j + 12

xj 

−12

s(q(t,x))dx . (6.16)

Defining the cell-average

Qj (t) = 1

xj 

 xj + 1

2

xj − 1

2

q(t,x)dx (6.17)

on the middle points (cell centers)

xj  =1

2 x

j − 12

+ xj + 1

2 , j  = 1, . . . , N   (6.18)

Equation (6.16) can be written as

Qj (t) = − 1

xj 

f 

q

t, xj + 1

2

− f 

q

t, xj − 1

2

+ 1

xj 

 xj + 1

2

xj − 1

2

s(q(t,x))dx .

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The approximation of the right hand side for j  = 1, . . . , N   leads to a system of 

ordinary differential equations (ODEs):

Qj (t ) =

− 1

xj 

f ∗

j + 12

− f ∗j − 1

2

+ sj 

(Q(·; t)) (6.19)

The right-hand side usually depends not only on one state Qj  but also on the states

of some neighboring cells.

The computation of f ∗j + 1

2

, f ∗j − 1

2

requires the solution of local Riemann problems.

This is carried out through the flux-difference splitting approach described in [1, 2].

This scheme was derived for the homogeneous flow Equations (6.7) for rectangularcross-sections in one space dimension and (6.1) in two space dimensions. A modi-

fication is given in [19, 32], in order to properly take into account the source terms

friction and bottom slope.

A suitable discretization of the dead-zone equations (6.9) and (6.10) was presented

in [13]. To avoid negative concentrations an ENO scheme was proposed for the

discretization of the convective term −ucx and a standard central finite difference

discretization for the diffusion term DLcxx . This ENO approach can be applied to

the 1D Saint–Venant Equations (6.6) for arbitrary cross-sections, too. It should bementioned that the 2nd order ENO and 2nd order flux-difference schemewith minmod

limiter [35] lead to nearly identical numerical results, if applied to Equation (6.7).

The idea of this ENO discretization is the reconstruction of the numerical flux

function f ∗j + 1

2

by a primitive function. The primitive function can be approximated

by polynominal interpolation. This interpolation is calculated via divided differences

and the set of points (stencil) included in the interpolation is chosen in order to get a

smooth polynominal.

6.3.3 Time Integration

The most attractive feature in MOL applications is the possibility to use high

quality and sophisticated integration schemes of high order for the semidiscretized

equations. Since the system of differential equations or DAEs generated via the

space-discretization process is usually stiff, implicit or semi-implicit time-integration

schemes are preferable. Rosenbrock–Wanner (ROW) methods are known to be effi-

cient for moderate accuracy requirements [38]. Due to their semi-implicit structurethey allow large time-steps, which are appropriate for the simulation of slowly vary-

ing flow problems. A ROW-method with stage-number s for the numerical solution

of a linear-implicit index 1 DAE system of the type

My = f (t, y) , t  ∈ [t 0, t 1] (6.20)

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with possible singular (n,n)-matrix M  and given consistent initial values y(t 0) is

defined by

y1 = y0 +s

i=1

bi Ki (6.21)

M  − hγ 

∂f 

∂y(t 0, y0)

Ki = hf 

t 0 + αi h, y0 +

i−1j =1

αij Kj 

+ γ i h2 ∂f 

∂t (t 0, y0)

+h

∂f 

∂y(t 0, y0)

i−1

j =1

γ ij Kj  (6.22)

with

αi =i−1j =1

αij , γ i =i

j =1

γ ij , γ ii = γ  (6.23)

and the coefficients γ , αij , γ ij , i = 1, . . . , s, j  = 1, . . . , i − 1 and weights bi ,

y1 being the approximation to the solution at time t 

+h with y(t)

=y0.

Unfortunately severe order-reductions can occur if Runge–Kutta or Rosenbrock 

methods are applied to semidiscretized PDEs [37, 20, 18]. This can be demonstrated

via the following example.

Let

u(x,t) = xφ(t) + (1 − x)ψ(t) φ, ψ ∈ C1[0, ∞) .

be the solution of the parabolic problem

ut  = uxx + f , x ∈ [0, 1] , t  ≥ 0

with f ( x , t ) = xφ + (1 − x)ψ and u(0, t) = ψ (t ), u(1, t) = φ(t), u(x, 0) =xφ(0) + (1 − x)ψ(0).

Semidiscretization on the equidistant mesh Xi = iN +1

, i = 1, . . . , N  with central

finite differences leads to

U  = AU  + B(t) + G(t) (6.24)

with U  = (U 1, . . . , U  N )t , U i = u(Xi , t), G = (G1, . . . , GN )

t , Gi (t) = Xi φ(t) +(1

−Xi )ψ(t), B(t)

=(N 

+1)2(ψ(t), 0, . . . , 0,φ(t))t  and matrix A defined by

A = −(N  + 1)2

2 −1

−1 2 −1

. . .. . .

. . .

−1 2 −1

−1 2

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Equation (6.24) is equivalent to

U  = A(U  − G) + G

and diagonalization of A = T −1T  yields the decoupled system

Y  = (Y  − G) + G  with Y  = T U  and G = T G

of Prothero–Robinson type [23]:

y = λ(y − g) + g , g(t) smooth and Reλ ≤ λ0 < 0 . (6.25)

This model has the exact solution y(t)

=g(t) for y(0)

=g(0) and for y(0)

=g(0)

with stiffness Reλ 0 the solution y(t) attains g(t) very quickly asymptotically.It is well known that many methods, if applied to the Prothero–Robinson model,

suffer from order-reduction. For the well-known ROW method RODAS [11], an

order reduction from theoretical order 4 to 1 can be observed [30]. RODAS is an

A-stable stiffly accurate embedded ROW method of order 4(3) with stage number

s = 6, s = 5. Scholz [28] derived additional order-conditions for ROW-methods to

overcome the order-reduction, and Ostermann and Roche [21] could show that these

additional conditions are sufficient to preserve the classical order of convergence on

certain classes of semidiscretized linear parabolic PDEs. In [30] a new coefficient set

for RODAS was derived in order to avoid order reduction phenomena in the context

of MOL. These coefficients are given in Table 6.1.

Table 6.1 Set of Modified Coefficients for RODAS, with βij  = αij  + γ ij 

γ  = 0.25 α21 = 0.75 β21 = 0.0α31 = 8.6120400814152190E − 2 β31 = −0.049392

b1 = β61 α32 = 0.1238795991858478 β32 = −0.014112b2

=β62 α41

=0.7749345355073236 β41

= −0.4820494693877561

b3 = β63 α42 = 0.1492651549508680 β42 = −0.1008795555555556b4 = β64 α43 = −0.2941996904581916 β43 = 0.9267290249433117b5 = β65 α51 = 5.308746682646142 β51 = −1.764437648774483b6 = γ α52 = 1.330892140037269 β52 = −0.4747565572063027

α53 = −5.374137811655562 β53 = 2.369691846915802

b1 = β51 α54 = −0.2655010110278497 β54 = 0.6195023590649829

b2 = β52 α61 = −1.764437648774483 β61 = −8.0368370789113464E − 2

b3 = β53 α62 = −0.4747565572063027 β62 = −5.6490613592447572E − 2

b4 = β54 α63 = 2.369691846915802 β63 = 0.4882856300427991

b5 = γ α64 = 0.6195023590649829 β64 = 0.5057162114816189α65 = 0.25 β65 = −0.1071428571428569

A disadvantage of ROW-methods, if applied to ODE systems y = f ( t , y ) of 

higher dimension, is the requirement of the exact Jacobian (∂f ∂y

) of the right-hand side

at every time-step. Therefore, the computation of the Jacobian and the solution of 

the linear equation systems are the main computational costs in case of integrating

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systems of large dimension. The code RODAS of Hairer and Wanner was modified

especially to take into account sparse non-banded Jacobians and to make efficient use

ofsparselinearalgebra software. This altered code, together with thenew coefficients,

is referred to as RODASP [30].The computation of the Jacobian is in most cases performed by finite differences.

In case of a full (n,n)-matrix this can be done by (n + 1) function evaluations of the

right-hand side f  with suitably chosen delta:

dy1=f(t,y)

for j=1 to n do begin

y(j)=y(j)+delta

dy=f(t,y)

for i=1 to n do begin

Jac(i,j)=(dy(i)-dy1(i))/deltaend

y(j)=y(j)-delta

end

In case of banded Jacobians with bandwidth m, the above algorithm is usually

modified so that it needs only m evaluations of f . The idea is to alter nm

components

of y at once before f  is evaluated. These components must be chosen so that two

or more nonzero entries in one row i never appear for two different components j.

This idea can be exploited directly for the computation of sparse Jacobians if the

sparsity structure is known [4]. Often it is easy to modify the subroutine defining

the right-hand side in order to determine the sparsity structure of the Jacobian in a

preprocessing step.

The linear algebra routines in RODASP allow the treatment of sparse matrices

and the solution of the linear equations with the preconditioned BI-CGSTAB algo-

rithm [36] or with the well-known method MA28 for the solution of unsymmetric

sparse systems [6, 5].

Another extension of RODASP was necessary in order to use it in combination

with the second-order, flux-difference splitting or ENO space-discretization schemes.Since the positions of the ENO interpolation stencil can change at every evaluation of 

the right-hand side f ( t , y ), the numerically computed Jacobian may not reflect (∂f ∂y

)

consistently. Therefore, at the beginning of each time-step the stencil positions have

to be chosen once and are fixed during all other right-hand side evaluations within

this time-step.

This feature is demonstrated by the 1D idealized dam-break problem. A 2000-m

long channel is assumed to be rectangular, horizontal, and frictionless. The reservoir

water (with depth hR) and the tail water (with depth hT ) are separated by a dam placed

in the middle (xM ) of the channel.The sudden and complete removal of the dam canbesimulated by the homogeneous

initial value problem (6.7) with

h(0, x) =

hR if x < xM 

hT  if x > xM 

u(0, x) = 0 .

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FIGURE 6.5

1D dam-break: setup.

By the theory of simple waves the exact depth h(t,x) is found to be a piecewise

smooth function on four varying intervals [34]:

h(t,x) =

h1 = hR  if  x ≤ −c1t 

h2 = 1

9g

2c1− x−xM 

t 

2if  −c1t < x < (u3−c3)t 

h3 if  (u3−c3)t  ≤ x ≤ ξ t 

h4 = hT   if  x > ξ t 

where ci = √ ghi (i = 1, . . . , 4 and g = 9.81 ms2 ).

Still a system of three nonlinear equations has to be solved for the unknowns u3

(velocity behind the shock), c3 (wave speed behind the shock), and ξ  (shock speed

of the flood wave). Introducing the abbreviation η = ξ c4

they read:

conservation of mass:c3c4

= 

12

 1 + 8η2 − 1

(6.26)

conservation of momentum:

u3

c4 = η − 

1

4η1 + 

1 + 8η2

(6.27)

u + 2c = const. in regions 2 and 3:u3c4

+ 2c3c4

= 2 c1c4

(6.28)

After inserting (6.26) and (6.27) into (6.28) we realize that η, and therefore the

whole solution, is implicitly dependent on the ratio of the initial depths, since c1c4

= hR

h

T  .

Table 6.2 summarizes a comparison of the different methods. The simulations stop

at t End = 60.0 s and the water depth is compared to the analytical solution.The abbreviations are

Roe(2): Roe’s second order flux-difference splitting scheme with minmod-

limiter. It is well known [17] that monotone upwind centered scheme for

conservation law (MUSCL) extrapolation with the minimod limiter is identical

to a second-order ENO-extrapolation.

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FIGURE 6.7

Dam-break problem with Roe(2)*, grid-size: x = 10.

fronts occur, a very fine mesh for the whole space interval [α, β] is necessary. In this

case adaptive meshes would be preferable. The mesh is adapted during the integration

in such a way that a fine resolution is obtained near the fronts and a coarse one in

regions of smooth solution components.

One has to distinguish between static and dynamic remeshing [8, 14]. In dynamic

remeshing the space-discretization points are considered to be time dependent and

they move with the solution. The disadvantage is the introduction of new unknowns

(the meshpoints) and the altered structure of the semidiscretized equations [22].

In static remeshing a new grid is fixed after one or m integration steps depending

on the actual solution behavior. The solution has to be interpolated from the old

mesh onto the new one and the integration procedure can be continued. Since everystatic remeshing step introduces a new system of equations with possibly different

dimension, the application of one-step methods is preferable.

In the following example, the application of a static remeshing strategy is demon-

strated. The advection dominated problem

ct  = −ucx + Dcxx t  ≥ 0 , x ∈ [α, β]with Peclet-number

P e

=

u

D

(β

−α)

=103

and typical constants α = 0, β = 30000, u = 1, D = 30 is treated. The left boundary

condition

cα(t ) = 10 exp

−0.001

(t  − 7500)2

t 

represents a wave coming from left into the computational domain [α, β]. The prob-

lem is integrated until t  = 60.000. At this time the wave has left the interval [α, β].

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The equidistribution strategy of [15], which is also implemented in the SPRINT-

software [3], has been used. By this strategy it is possible to construct a locally

bounded mesh α = X0 < · · · < XN  = β with respect to a bound K ≥ 1:

1

K≤ Xi+1 − Xi

Xi − Xi−1≤ K .

Moreover, the mesh is sub-equidistant with respect to a mesh-function m(x) and a

constant c > 0 with

N c ≥ b

a

m(x)dx and

 Xi+1

Xi

m(x)dx ≤ c for i = 0, . . . , N  − 1 .

The mesh-function was chosen as

m(x) = 

σ  + c2xx   (6.29)

By the parameter σ , the maximum possible mesh-size can be controlled.

Figure 6.8 shows a typical adaptive grid during simulation. Finite differences have

been applied in this example for space discretization. In [26] it has been shown that

this type of problem can benefit from adaptive grid strategies. Fixing the CPU-time,

the maximum error is halved; or describing an equal error, the number of grid points

is smaller for the adaptive strategy. For larger Peclet-numbers the improvements are

increasing.

FIGURE 6.8

Adaptive grid.

6.4.1 Extension to 2D Problems

The equidistribution strategy can easily be extended to 2D river flow problems. In a

first step, a basic mesh for the 2D domain must be constructed. Figure 6.9 shows such

a coarse mesh for a stretch of the lower River Saar. This mesh has been constructed

orthogonal to the main flow direction by a linear affine interpolation [39].

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FIGURE 6.9

Basic 2D mesh.

In the next step, one preferred direction along the river course is chosen and

parametrizations along and across this direction are defined (cf. Figure 6.10).

Parametrizations along the river

s −→ (xm(s),ym(s))t  , s ∈ [0, 1]⇒ s −→ (xl (s), yl (s))t  , s −→ (xr (s), yr (s))t 

and for all s ∈ [0, 1] across the river:

t  −→ (x(s,t),y(s,t))t , t  ∈ [0, 1]x(s,t)

y(s,t)

=

xl (s)

yl (s)

+ t 

xr (s) − xl (s)

yr (s) − yl (s)

FIGURE 6.10

Parametrization of space coordinates.

Now, a mesh function m(s) along the river course can be defined by

m(s) = 1

0

c(x(s,t),y(s,t))dt 

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with a certain function c(x,y), e.g., c(x,y) =√ 

u2 + v2 (norm of velocities).

This mesh function is equidistributed and new mesh points si , i = 1, . . . , n in s

direction can be calculated.

In the next step, functions mi (t) are defined for all si , i = 2, . . . , n − 1:

mi (t) = si+1

si−1

c(x(s,t),y(s,t))ds

The equidistribution of mi (t) leads to new mesh points t ij , j  = 1, . . . , m across the

river.

If necessary, a more smooth behavior of the mesh points and less distorted angles

can be obtained through the modified mesh functions

mi (t) = 

si+k

si−k

c(x(s,t),y(s,t))ds, k > 1

Figure 6.11 shows the new resulting mesh. This mesh has been adapted to the bottom

elevation in t  direction with the choice c(x,y) = bmax − b(x,y), bmax being the

maximum bottom elevation in the corresponding cross-section. In Figure 6.11 the

course of the river bed within the flood plane can be seen.

FIGURE 6.11

Adaptive mesh.

Finally, an interpolation of the solution from the old onto the new mesh has to be

performed. Remember that in the finite volume approach the unknowns Qj  are mean

values on the mesh cells j : Qj  = 1|j |

 j 

q dA.

The interpolated solution can be calculated via

Qj new= 1

|j new|

i

|i ∩ j new|Qi

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This approach has been tested for the Molenkamp problem (2D convection):

ct  = ucx + vcy , x ∈ [−1, 1], y ∈ [−1, 1]

with u = −2πy, v = 2π x.

An analytical solution is

c(x,y,t) = 0.014r2

, with r = 

(x + 1

2cos 2π t)2 +

y + 1

2sin 2π t 

2

Initial condition and boundary conditions are chosen according to this solution.

Table 6.3 compares the fixed and adaptive solutions after one revolution (t end = 1)

on different meshes, being the maximum absolute error. For the time integration inthis non-stiff example the explicit scheme DOPRI5 [10] has been applied. After every

5 to 20 time steps a remeshing has been done. It can be concluded that the adaptive

approach leads to an improved accuracy, but CPU time is increased extremly due to

the 2D interpolation. For a final maximum absolute error , the computation times

are nearly equivalent, see the results for = 0.4 in row one and = 0.39 in row

two. Thus, this proposed adaptive method does not really pay off in 2D problems.

Nevertheless, it is very useful for mesh construction. The initial adaptive 31 × 31

mesh is shown in Figure 6.12.

Table 6.3 Numerical Results for Molenkamp Problem

31 × 31 41 × 41 51 × 51 61 × 61

Fixed 0.62 0.47 0.40 0.29

CPU 19 45 87 153

Adaptive 0.39 0.25 0.19

CPU 101 233 488

6.5 Applications

The aim of the model WAFOS is the forecasting of water levels at important gauges

along the river during low water for navigation and during floods for flood warning.

Usually, a daily 48-h forecast run is performed on the basis of measured water levelsup to 7.00 am. During floods, the model is operated up to three times a day. In this

case the results up to a forecast time of 24 h are disseminated for 18 main gauges

on River Rhine downstream of Karlsruhe/Maxau.  Figure 6.13 shows an example of 

forecast results for gauging station Koblenz during a medium flood in 1999.

Since the gauging station Koblenz is located 800 m upstream of the junction with

the Moselle, severe backwater effects occur. Therefore, a coupled hydrodynamic

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FIGURE 6.12

Initial adaptive 31

×31 mesh for Molenkamp problem.

FIGURE 6.13

Forecast results for gauging station Koblenz/Rhine.

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1000 m

125.94

127.90

130.34

132.79

135.23

137.68

140.12

142.57

145.01

147.46

149.90

FIGURE 6.15

Bottom elevations (adaptive 296 × 72 mesh).

Wanner method for the numerical solution of the semi-discretized PDEs is a robust

and reliable scheme and well suited for a daily use simulation software.

Acknowledgment

The authors would like to thank Michael Hilden, Adrian Q.T. Ngo, and Silke

Rademacher for helpful contributions and cooperation and Dr. Klaus Wilke for his

support.

References

[1] F. Alcrudo, P. Garcia-Navarro, and J.-M. Saviron, Flux-difference splitting for

1D open channel flow equations, Int. J. f. Num. Meth. Fluids, 14, (1992), 1009–

1018.

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on Flood Forecasting, Czech Hydrometeorological Institute, 108–117 Prague,

1998.

[17] R.J. LeVeque, Numerical methods for conservation laws, Lectures in Mathe-

matics, Birkhäuser, Zürich, 1992.

[18] Ch. Lubich and A. Ostermann, Runge–Kutta methods for parabolic equations

and convolution quadrature, Math. Comp., 60, (1992), 105–131.

[19] Q.T. Ngo, Numerical simulation of river flow problems based on a finite volume

model, Diploma thesis, Univ. Kaiserslautern, 1999.

[20] A. Ostermann and M. Roche, Runge–Kutta methods for partial differential

equations and fractional orders of convergence, Math. Comp., 59, (1992), 403–

420.[21] A. Ostermann and M. Roche, Rosenbrock methods for partial differential equa-

tions and fractional orders of convergence, SIAM J. Numer. Anal., 30, (1993),

1084–1098.

[22] L.R. Petzold, Observations on an adaptive moving grid method for one-

dimensional systems of partial differential equations, Applied Numer. Math.,

3, (1987), 347–360.

[23] A. Prothero and A. Robinson, The stability and accuracy of one-step methods,

 Math. Comp., 28, (1974), 145–162.

[24] P. Rentrop, M. Hilden, and G. Steinebach, Wissenschaftliches Rechnen, Der 

 Ingeniuer in der Wasser- und Schifffahrtsverwaltung, 19, (1999), 19–23.

[25] P. Rentrop and G. Steinebach, Model and numerical techniques for the alarm

system of river Rhine, Surveys Math. Industry, 6, (1997), 245–265.

[26] P. Rentrop and G. Steinebach, A method of lines approach for river alarm

systems, ECMI Progress in Industrial Mathematics at ECMI’96, Brons, M.,

Bendsoe, M.P., Sorensen, M.P., eds., 12–19, Teubner Stuttgart, 1997.

[27] W.E. Schiesser, The Numerical Methods of Lines, Academic Press, San Diego,

CA, 1991.

[28] S. Scholz, Order barriers for the B-convergence of ROW methods, Computing,

41, (1989), 219–235.

[29] M. Spreafico and A. van Mazijk, Alarmmodell Rhein, Ein Modell für die oper-

ationelle Vorhersage des Transportes von Schadstoffen im Rhein, KHR-Bericht 

 Nr. I-12, Lelystad, 1993.

[30] G. Steinebach, Order-reduction of ROW-methods for DAEsand method of lines

applications, Preprint-Nr. 1741, FB Mathematik, TH Darmstadt, 1995.

[31] G. Steinebach, Using hydrodynamic models in forecast systems for large rivers,

Proc.Advances inHydro-Scienceand-Engineering,Vol.3 incl.CD-ROM,Holz,

K.P., Bechteler, W., Wang, S.S.Y., Kawahara, M., eds., Cottbus, 1998.

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[32] G. Steinebach and A.Q.T. Ngo, A method of lines flux-difference splitting finite

volume approach for 1D and 2D river flow problems, to appear in Godunov

 Methods: Theory and Applications, E.F. Toro, ed., Kluwer Academic/Plenum

Publishers, 2001.

[33] G. Steinebach and K. Wilke, Flood forecasting and warning on the River Rhine,

Water and Environmental Management, J. CIWEM, 14, (2000), 39–44.

[34] J.J. Stoker, Water Waves, the Mathematical Theory with Applications, Inter-

science Publishers, New York, 1957.

[35] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics,

Springer, Berlin, Heidelberg, 1999.

[36] H.A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comp.,

13, (1992), 631–644.

[37] J.G. Verwer, Convergence and order reduction of diagonally implicit Runge–

Kutta schemes in the method of lines, in Griffiths, Watson: Numerical Analysis,

Pitman Research Notes in Mathematics, 220–237, 1986.

[38] J.G. Verwer, W.H. Hundsdorfer, and J.G. Blom, Numerical time integration for

air pollution models, Modeling, Analysis and Simulation Report MAS-R9825,

58 p., CWI Amsterdam, 1998.

[39] C.B. Vreugdenhil, Numerical methods for shallow-water flow, Kluwer Acad.

Pub., Dordrecht, 1994.

[40] K. Wilke, Mehrkanalfiltermodell (MKF), in Beschreibung hydrologischer 

Vorhersagemodelle im Rheineinzugsgebiet, Bericht I-7 der KHR, Lelystad, 71–

85, 1988.

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Chapter 7 

 An Adaptive Mesh Algorithm for FreeSurface Flows in General Geometries

Mark Sussman1

7.1 Introduction

In this chapter we present an adaptive method for computing incompressible free

surface flows in general geometries. An example of two flows that we consider are

(1) flows in a 3D jetting device (Figure 7.11) and (2) ship waves (Figure 7.9). Our

computations are done on an adaptive grid as described by Berger and Colella [6]

and Almgren et al. [1]. The free surface separating the gas and liquid is modeled

using “embedded boundary” techniques; specifically, a coupled level set and volume

of fluid method is used [50]. Our method for modeling the free surface allows for

the arbitrary merge and break-up of fluid mass while maintaining excellent mass

conservation. An “embedded boundary” (a.k.a. Cartesian grid [20]) method is also

used to represent irregular geometries (e.g., ship hull or jetting housing). In theprocess of describing our methods for modeling the free surface and geometry, we

also present a new (easy) way for enforcing the contact angle boundary condition at

points where the free surface meets the geometry.

7.1.1 Overview: Adaptive Gridding

For the problems we consider, dynamic adaptive grid refinement is important.

The error is largest in regions near the free surface. A finer mesh is needed at the

free surface more than elsewhere. There are quite a few numerical techniques for

implementing adaptivity. In the finite element framework, the reader is referred to

the following works [47, 37, 19, 9, 35, 63, 30, 36]. In the finite difference framework 

1Work supported in part by NSF# DMS 97-06847, DOE (MICS) program contract DE-AC03-76SF00098,

DOE (MICS) program contract DE-FG03-95ER25271, and an ASCI grant from the Los Alamos National

Laboratory.

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(i.e., uniform rectangular mesh), the reader is referred to the following dynamic

adaptive grid methods [6, 1, 48, 62, 34]. Other methods that allow one to add grid

resolution where needed are so-called Overset-grid methods [13, 18].

Inourwork, weadopt thefinite-difference-based adaptive grid techniques describedin [6] and extended to incompressible flows in [1]. The idea behind these methods

is that the basic numerical methodology used for a single rectangular mesh should

be unchanged when generalizing to a collection of rectangular meshes with differing

resolutions. The only additional logic added to the base numerical algorithm is to

be able to handle boundary conditions at coarse-fine grid interfaces or fine-fine grid

interfaces. Another advantage to the adaptive grid techniques that we adopt is that

these techniques are naturally parallelizable. Each rectangular grid can be assigned

a different processor.

7.1.2 Overview: Free Surface Model

There are two classes of free surface algorithms commonly used for incompressible

two-phase flow problems: (1) body-fitted or Lagrangian techniques and(2) embedded

boundary techniques.

In body-fitted/Lagrangian techniques [11, 10, 29, 61, 60], the computational grid is

aligned with the free surface at all times. These methods are generally more accurate

than their embedded boundary counterparts and also more efficient. Unfortunately,

these methods will break down when the free surface develops a change in topologyunless special measures are taken [51]. Also, there is a regridding issue as the free-

surface deforms.

In embedded boundary techniques [43, 52, 59, 12, 46, 23, 44, 58, 17, 16, 45, 27, 31,

26, 25] the free surface is allowed to cut through the computational grid. The compu-

tational grid remains fixed while the free surface deforms arbitrarily. These methods

are typically more robust and easier to program than their body-fitted/Lagrangian

counterparts. On the other hand, one generally cannot achieve higher than first-order

accuracy using embedded boundary techniques for the free surface.

In our work we adopt the “coupled level set volume of fluid” method describedin [50]. This method falls in the category of an “embedded boundary” technique.

The free surface cuts through the computational grid. The free surface is represented

“implicitly” by two field variables: (1) the level set function φ (x, t), positive in liquid

and negative in gas, (2) the volume of fluid function F (x, t), 1 in liquid, 0 in gas, and

0 < F < 1 in partially filled computational elements.

Remarks:

1. Front tracking approaches [59, 58] are generally more accurate approaches

for representing the embedded free surface than level set [52] or volume-of-fluid [12] methods; but front tracking methods are complicated to implement

for 3D problems with multiple changes in topology (e.g.,wavesloshing, droplet

break-up).

2. In our work we solve for the flow in both the liquid and gas. Some methods

(e.g., [16]) solve for the liquid only and assume pressure is constant in the gas.

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Methods that solve in the liquid only are generally more efficient, especially if 

the flow is comprised of a relatively small portion of liquid.

7.1.3 Overview: Modeling Flows in General Geometries

As with algorithms for free surfaces, algorithms for flows in general geometries

fall into two categories: (1) body-fitted and (2) embedded boundary.

In body-fitted techniques (structured, unstructured, mapped grids) [55, 14, 7, 32,

38, 5, 41, 56], the computational grid is aligned with the geometry. These methods are

generally more accurate than their embedded boundary counterparts and also more

efficient. A drawback with body-fitted techniques for flows in general geometries is

that one has to generate an appropriate grid and design a numerical method whichoperates on non-uniform/mapped grids.

In embedded boundary techniques (a.k.a. Cartesian grid methods) [42, 24, 33, 57,

2], the general geometry cuts through the computational grid. This allows one to use

algorithms designed for fixed rectangular grids with little modification.

In our work, the irregular boundary (e.g., ship hull or jetting device housing)

is represented as the zero level set of a second  level set function ψ along with the

corresponding area fractions A andvolume fractions V . ψ ispositive in the activeflow

region and negative elsewhere. V  = 1 for computational elements fully containedwithin the active flow region and V  = 0 for computational elements fully outside the

active flow region. The representation of irregular boundaries via area fractions and

volume fractions has been used previously in the following work for incompressible

flows [2, 57].

7.2 Governing Equations

We assume that both the liquid and the gas are incompressible, immiscible fluids.

The equations for both the liquid and the gas have the form

U t  + ∇ · (UU ) = −∇ p

ρ+

µU 

ρ− G. (7.1)

∇ · U  = 0

The quantities ρ and µ in (7.1) represent the values of liquid or gas depending on

what fluid one is in. The free surface boundary conditions are as follows:

U g = U 

l2µl Dl − 2µgDg

· n =

pl − pg + γ κ

n . (7.2)

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n is the outward normal drawn from the gas into the liquid. κ is the local mean

curvature of the free surface. D is the rate of deformation tension,

D =

1

2

∇ U  + ∇ U T 

.

We shall enforce the no-slip boundary conditions at solid walls,

U  = 0 . (7.3)

Also, at solid walls, we enforce a contact angle boundary condition,

n · nwall = cos(θ) , (7.4)

where θ  is a user-defined contact angle and nwall is the outward normal drawn fromthe active flow region into the geometry region.

The explicit enforcement of the free surface boundary condition (7.2) can be com-

plicated in 3D; especially for interfaces that can merge or break. Instead of solving

in the gas and liquid separately, and then coupling the solutions at the free surface,

we solve the following equations for both the gas and the liquid:

U t  = −∇ · (UU ) −∇ p

ρ(φ)+

∇ · 2µ(φ)D

ρ(φ)−

γκ(φ)∇ H(φ)

ρ(φ)− G . (7.5)

∇ · U  = 0

φt  + U  · ∇ φ = 0 (7.6)

ρ(φ) ≡ ρl H(φ) + ρg (1 − H(φ))

µ(φ) ≡ µl H(φ) + µg (1 − H(φ))

H(φ) =

1 φ > 0

0 otherwise

κ(φ) ≡ ∇ ·∇ φ

|∇ φ|The level set function φ is defined to be positive in the liquid and negative in the

gas. The motion of the free surface is determined from the level set equation (7.6).

The level set equation tells us that φ remains constant on particle paths. In other

words, if the zero level set of  φ coincides with the free surface, then solutions at

a later time will also have the zero level set of φ coinciding with the free surface.

It has been shown by Chang et al. [15] that weak solutions of (7.5) satisfy the free

surface boundary conditions (7.2). We never have to explicitly enforce thefree surface

boundary conditions. They are implicitly enforced through the use of the Heaviside

function H(φ).

7.2.1 Projection Method

In order to solve (7.5), we use a variable density projection method [4] which is a

generalization of the constant density projection method presented by [3]. First, one

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can rewrite (7.5) in the following form:

W d  + ∇ p/ρ = W  (7.7)

where W d  represents U t  and W  represents,

W  = −∇ · (UU ) +∇ · 2µ(φ)D

ρ(φ)−

γ κ∇ H(φ)

ρ(φ)− G.

After taking the divergence of both sides of (7.7), we use the continuity equation in

order to set ∇ · W d  = 0, thus resulting in the following equation for the pressure field:

∇ ·∇ p

ρ

= ∇ · W . (7.8)

In order to impose a no-outflow condition at solid walls, one has the following Neu-

mann boundary condition on p:

∇ p

ρ· nwall = W  · nwall.

Once the pressure field p is determined from (7.8), one can then update W d  as

W d  = W  − ∇ p/ρ .

We shall denote the projection operator as:

W d  = P ρ ( W ) .

The resulting equations to be solved now, when written in terms of the projection

operator, are

U t  = P ρ

−∇ · (UU ) +

∇ · 2µ(φ)D

ρ(φ)−

γ κ∇ H(φ)

ρ(φ)− G

(7.9)

φt  + U  · ∇ φ = 0 .

7.3 Discretization

We discretize (7.9) on a fixed rectangular grid. The free surface and geometry are

embedded within the grid. The free surface is represented as the zero level set of asmooth function φ. The geometry is represented as the zero level set of a smooth

function ψ . Important to our numerical scheme is the volume fraction F  of liquid

in each computational element. For each computational element, ij , the volume

fraction of liquid is defined as:

F ij  ≡1

|ij |

 ij 

H(φ)d x .

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Also important to our numerical scheme is the geometry volume fraction V  of ij ,

and the geometry area fraction A of i+1/2,j ,

V ij  ≡

1

|ij | 

ij H(ψ)d x .

Ai+1/2,j  ≡1

|i+1/2,j |

 i+1/2,j 

H(ψ)d x .

i+1/2,j  represents the left face of a computational element; similar definitions apply

to i−1/2,j , i,j +1/2, i,j −1/2.

The state variables uij , φij , and F ij  are stored at the center of each computational

grid cell. The pressure pi+1/2,j +1/2 is stored at the cell corners (nodes).

A simple first-order discretization is as follows:

1. Given φn, F n, U n

2a. Set U n = 0 in computational cells where V ij  = 0, i.e., velocity satisfies no-

slip conditions on geometry walls and velocity is identically zero within the

geometry. This is a first-order boundary condition; for an example of higher-

order Cartesian Grid discretizations, see [33, 20].

2b. Extend φn into regions where V ij  < 1. The extension procedure “implicitly”

enforces the contact angle boundary condition

n · nwall = cos(θ) .

The extension procedure is described in Section 7.5.2.

3. Form

V n = −[∇  (UU )]n +

∇ · (2µ(φn)Dn)

ρ(φn)−

γκ(φn)∇ H (φn)

ρ(φn)− G

4. Update the position of the free surface using the “coupled level set volume-of-

fluid” (CLS) method (see Section 7.4),

φn+1 = φn − t [∇ · (uMAC φ)]n

F n+1 = F n − t [∇ · (uMAC F )]n

Remark: modifications to the (CLS) method for general geometries are pro-

vided in Section 7.5.3.

5. Update the velocity (pressure solve)

U n+1 = U 

n + tP ρ(φn)(V n) (7.10)

6. Reinitialize φn+1 using current values for φn+1 and F n+1 (maintain φn+1 as

the signed distance from the zero level set of φn+1).

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Remarks:

• The nonlinear term [∇ (UU )]n is discretized using a second-order, slope-limited

predictor corrector method described in [48].

• For the sake of readability, we describe the first order in time method above.

In practice we employ the second-order “Crank-Nicolson” time discretization

described by Bell et al. [3] and also implemented in [48].

• The step that takes the most time is the projection step (7.10). In the projection

step, we solve the following discretized equation for p,

∇ ·

1

ρ(φn) ∇ p = ∇ · V n

, (7.11)

subject to the boundary conditions

∇ p

ρ(φn)· nwall = V 

n · nwall .

Details of how we enforce the no-flow condition at solid walls are given in

Section 7.5.1.2 below.

Inorder tosolve the resulting linearsystem, weuse the multigrid preconditioned

conjugate gradient method [54].

• We use time-step constraints due to the CFL condition, viscous terms, and

surface-tension terms. For jetting problems, it is the surface-tension, time-step

constraint which is most restrictive.

7.3.1 Thickness of the Interface

In the discretization of (7.5) we replace H(φ) with H  (φ) where H  (φ) is defined

as

H  (φ) =

0 φ < −12

1 + φ

+ 1

πsin(π

φ

)

|φ| ≤

1 φ >

For most of our computations, = 3x. For a few 3D problems (see remark in

Section 7.7.1.1) we set = 4x. Without smoothing (i.e., = 0), our method yields

oscillatory results, probably due to the fact that with zero thickness the tangentialvelocity jumps sharply across the free surface (high Reynolds number flows).

Because we give the interface a thickness , we find it necessary to maintain the

level set function φ as a signed distance function. Otherwise, one would not have a

uniform thickness. Over time, the nonzero level sets can stack up in some regions

and spread apart in others. In [52], a comparison is given between computations with

and without reinitialization for a rising steady-gas-bubble problem; it was shown that

reinitialization is needed in order to preserve the steady solution.

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As a remark, in work by Sussman and Smereka [51] the level set method was

compared to the boundary integral method for bubble and drop problems. The two

methods compared very well despite the fact that the level-set method gives the

interface a thickness whereas the boundary-integral method considers the interfaceas sharp.

7.4 Coupled Level Set Volume of Fluid Advection Algorithm

In this section, we describe the 2D coupled level set and volume of fluid (CLS)

algorithm for representing the free surface. For more details, e.g., axisymmetric and3D implementations, see [50]. In the CLS algorithm, the position of the interface is

updated through the level-set equation and volume-of-fluid equation,

φt  + ∇ ·U 

MAC φ

= 0

F t  + ∇ ·U 

MAC F 

= 0 .

In order to implement the CLS algorithm, we are given a discretely divergence-free

velocity field uMAC defined on the cell faces (MAC grid),

ui+ 1

2,j 

− ui− 1

2,j 

x+

vi,j + 1

2− v

i,j − 12

y= 0 . (7.12)

Given φnij , F nij , andU MAC , we usea “coupled”second-order, conservative-operator

split advection scheme in order to find φn+1ij  and F n+1

ij  . The 2D operator split algo-

rithm for a general scalar s follows as

sij  =sn

ij  + t x

G

i− 12

,j − G

i+ 12

,j 

1 − t x

u

i+ 12

,j − u

i− 12

,j 

(7.13)

sn+1ij  = sij  +

t 

y

G

i,j − 12

− Gi,j + 1

2

+ sij 

v

i,j + 12

− vi,j − 1

2

, (7.14)

where Gi+ 1

2,j 

= si+ 1

2,j 

ui+ 1

2,j 

denotes the flux of s across the right edgeof the (i,j)th

cell and Gi,j + 12 = si,j + 1

2 vi,j + 12 denotes the flux across the top edge of the (i,j)th

cell. The operations (7.13) and (7.14) represent the case when one has the “x-sweep”

followed by the “y-sweep.” After every time step the order is reversed; “y-sweep”

(done implicitly) followed by the “x-sweep” (done explicitly).

The scalar flux si+ 1

2,j  is computed differently depending on whether s represents

the level-set function φ or the volume fraction F .

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For the case when s represents the level-set function φ, we have the following

representation for si+ 1

2,j  (u

i+ 12

,j > 0):

si+ 1

2,j 

= snij  + x

2(Dx s)n

ij  + t 2

−u

i+ 12

,j  (Dx s)nij 

where

(Dx s)nij  ≡

sni+1,j  − sn

i−1,j 

x.

The above discretization is motivated by the second-order, predictor-corrector method

described in [3] and the references therein.

For the case when s represents the volume fraction F  we have the following rep-

resentation for si+ 1

2,j 

(ui+ 1

2,j 

> 0):

si+ 1

2,j  =

  H (φ

n,Rij  (x, y))d

ui+ 1

2,j 

ty(7.15)

where

≡ (x,y)|xi+

1

2

− ui+

1

2 ,j 

t  ≤ x ≤ xi+

1

2

and yj −

1

2

≤ y ≤ yj +

1

2

The integral in (7.15) is evaluated by finding the volume cut out of the region of 

integration by the line represented by the zero level set of φn,Rij  .

The term φn,Rij  (x,y) found in (7.15) represents the linear reconstruction of the

interface in cell (i,j). In other words, φn,Rij  (x,y) has the form

φn,Rij  (x,y) = aij (x − xi ) + bij (y − yj ) + cij  . (7.16)

A simple choice for the coefficients aij  and bij  is as follows:

aij  =1

2x(φi+1,j  − φi−1,j ) (7.17)

bij  =1

2y(φi,j +1 − φi,j −1) . (7.18)

The intercept cij  is determined so that the line represented by the zero level set

of (7.16) cuts out the same volume in cell (i,j) as specified by F n

ij . In other words,the following equation is solved for cij :

 

H (aij (x − xi ) + bij (y − yj ) + cij )d

xy= F nij 

where

≡

(x,y)|xi− 1

2≤ x ≤ x

i+ 12

and yj − 1

2≤ y ≤ y

j + 12

.

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After φn+1 and F n+1 havebeen updated according to (7.13) and (7.14) we “couple”

the level-set function to the volume fractions as a part of the level-set reinitialization

step. The level-set reinitialization step replaces the current value of φn+1 with the

exact distance to the VOF reconstructed interface. At the same time, the VOF re-constructed interface uses the current value of φn+1 to determine the slopes of the

piecewise linear reconstructed interface.

Remarks:

• The distance is only needed in a tube of K cells wide K = /x +2, therefore,

we can use “brute force” techniques for finding the exact distance. See [50] for

details.

• During the reinitialization step we truncate the volume fractions to be 0 or 1

if |φ| > x. Although we truncate the volume fractions, we still observe thatmass is conserved to within a fraction of a percent for our test problems.

7.5 Discretization in General Geometries

The discretization of the following items need additional explanation in general

geometries:

1. Projection step

2. Surface tension (contact angle boundary conditions)

3. CLS advection

7.5.1 Projection Step in General Geometries

7.5.1.1 MAC Project

In order to construct the advective “MAC” velocities (7.12) located at cell face cen-

troids (see [1] for further details of the “MAC” projection step), a “MAC” projection

step is needed.

In the MAC projection step, we solve the following discretized equation for p:

∇ ·1

ρ(φn)∇ p = ∇ · V n , (7.19)

subject to the boundary conditions

∇ p

ρ(φn)· nwall = V 

n · nwall . (7.20)

In order to discretely enforce the boundary conditions (7.20) at the geometry surface,

we use a finite-volume approach for discretizing (7.19).

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and

∇ · V n ≈ 1V ij xy

Ai+1/2,j y

ui+1/2,j  −

Ai−1/2,j y

ui−1/2,j 

+

Ai,j +1/2x

vi,j +1/2 −

Ai,j −1/2x

vi,j −1/2 − Lwallij  V 

n,wallij  · nwall

.

Dueto the no-flow condition(7.20), the terms Lwallij  (∇ p/ρ)wall

ij  ·nwall and Lwallij  V 

n,wallij  ·

nwall cancel each other. The resulting discretization for p is:

Ai+1/2,j y(px /ρ)i+1/2,j  − Ai−1/2,j y(px /ρ)i−1/2,j 

+Ai,j +1/2x(py /ρ)i,j +1/2 − Ai,j −1/2x(py /ρ)i,j −1/2

= (Ai+1/2,j 

y)ui+1/2,j 

− (Ai−1/2,j 

y)ui−1/2,j 

+(Ai,j +1/2x)vi,j +1/2 − (Ai,j −1/2x)vi,j −1/2

where, for example, (px )i+1/2,j  is discretized as

pi+1,j  − pi,j 

x.

7.5.1.2 Nodal Projection

The “nodal” projection step solves (7.11) for pi+1/2,j +1/2 subject to the followingboundary conditions at the embedded boundary:

∇ p

ρ(φn)· nwall = V 

n · nwall. (7.23)

The following modification of (7.11) implicitly enforces (7.23),

∇ ·1

ρ(φn)H(ψ)∇ p = ∇ · H(ψ)V n , (7.24)

where H  is the Heaviside function. In other words, weak solutions of (7.24) auto-

matically satisfy (7.23). We solve (7.24) in a fixed rectangular domain that contains

the embedded geometry (the zero level set of ψ).

In order to discretize (7.24), we modify the standard discretization of the following

pressure equation:

∇ ·1

ρ(φn)∇ p = ∇ · V n ,

by replacing 1ρ(φn

ij )with

V ij 

ρ(φnij )

and by replacing V nij  with V ij V 

nij .

Remark:

Our discretization of (7.24) is only first-order accurate at cells that have a partial

geometry volume fraction 0 < V ij  < 1. For possible higher order discretizations, we

refer the interested reader to [33, 20].

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7.5.2 Contact-Angle Boundary Condition in General Geometries

The contact-angle boundary condition at solid walls is given by (7.4). In terms of 

φ and ψ , (7.4) becoc∇ φ

|∇ φ|·

−∇ ψ

|∇ ψ |= cos(θ) .

In Figure 7.2, we show a diagram of how the contact angle θ  is defined in terms of 

how the free surface intersects the geometry surface.

FIGURE 7.2

Diagram of gas/liquid interface meeting at the solid. The dashed line represents

the imaginary interface created through the level-set extension procedure.

The “extension” equation has the form of an advection equation:

φτ  + uextend · ∇ φ = 0 ψ < 0 (7.25)

In regions where ψ ≥ 0, φ is left unchanged.

For a 90◦ contact angle, we have

uextend = −∇ ψ

|∇ ψ |.

In other words, information propagates normal to the geometry surface.

For contact angles different from 90◦, the following procedure is taken to finduextend:

n ≡∇ φ

|∇ φ|

nwall ≡ −∇ ψ

|∇ ψ |

n1 ≡ −n × nwall

|n × nwall|

n2 ≡ −n1 × nwall

|n1 × nwall|c ≡ n · n2

uextend =

nwall−cot(π−θ )n2

|nwall−cot(π−θ )n2| if c < 0

nwall+cot(π−θ )n2

|nwall+cot(π−θ )n2| if c > 0

nwall if c = 0

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Remarks:

• In 3D, the contact line (CL) is the 2D curve which represents the intersection

of the free surface with the geometry surface. The vector n2 is orthogonal to

the CL and lies in the tangent plane of the geometry surface.

• Since both φ and ψ are defined within a narrow band of the zero level set of φ,

we can also define uextend within a narrow band of the free surface.

• We use a first-order upwind procedure for solving (7.25). The direction of 

upwinding is determined from the extension velocity uextend. We solve (7.25)

for τ  = 0 . . . .

• For viscous flows, there is a conflict between the no-slip condition (7.3) and theidea of a moving contact line. See [28, 40, 21] and the references therein for a

discussion of this issue. We haveperformed numerical studies for axisymmetric

oil spreading in water under ice [53] with good agreement with experiments.

In the future, we wish to experiment with appropriate slip-boundary conditions

near the contact line.

7.5.3 CLS Advection in General Geometries

For computational elements which contain only air and/or water, i.e., V ij  = 1,the CLS advection algorithm as described in Section 7.4 remains unchanged. For

computational elements in which 0 < V ij  < 1, we use the extension procedure

described in Section 7.5.2 in order to initialize the level-set function φ and the volume

fraction F  in partial elements.

Remarks:

1. Since we only discretize the CLS advection step in full cells, we avoid stringent

CFL conditions that exist in very small partial cells.

2. The discretization of the CLS advection step is not conservative. See [42, 20]

for conservative “finite-volume-based” alternatives.

7.6 Adaptive Mesh Refinement

We describe the extension of the single-grid algorithm (Section 7.3) to an adap-

tive hierarchy of nested rectangular grids. For general references on adaptive mesh

refinement (AMR) we refer the reader to [8, 6, 39, 1]. The ideal of AMR is that the

solution procedure for a fixed uniform computational grid should remain unchanged.

Adaptivity is achieved by dynamically overlaying successively finer grids in order

to increase resolution of the free surface. The main modification to the single-grid

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algorithm would be to supply boundary conditions at points where coarse grids and

fine grids meet.

In Figure 7.3 we show an example of the grid structure used in AMR. The grid

hierarchy is composed of different levels of refinement ranging from coarsest, = 0,to finest, = max. The coarsest level, = 0, covers the whole computational domain

while successively higher levels, + 1, lie on top of the level underneath them, level

.

FIGURE 7.3

Diagram of grid structure used in adaptive mesh refinement (AMR). In this

example, there are 3 levels. Level 0 has one 16 × 16 grid. Level 1 has two grids:

a 16 × 16 grid and a 8 × 14 grid. Level 2 also has two grids: a 16 × 20 grid and

a 16 × 12 grid. The refinement ratio between levels in this example is 2.

7.6.1 Time-Stepping Procedure for Adaptive Mesh Refinement

We use a “no-sub-cycling” time stepping procedure. In other words, the time

step used on the finest level is the same as on all other levels. The details of our

implementation can be found in [48]. An outline of our adaptive algorithm is as

follows:

1. Given φn, F n, U n on coarse and fine levels.

2a. For coarse and fine levels, set un = 0 in computational cells where V ij  = 0.

2b. Extend φn into regions where V ij  = 0. Repeat on all AMR levels (coarse and

fine grids).

3. Repeat on coarse and then finer level(s):

V n = −[∇ (UU )]n +

∇ · 2µ(φnDn)

ρ(φn)−

γκ(φn)∇ H (φn)

ρ(φn)− G .

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FIGURE 7.4

Axisymmetric jetting of ink. ρw/ρa = 816, µw/µa = 64. Effective fine grid

resolution is 64 × 1024.

-500000

0

500000

1e+06

1.5e+06

2e+06

2.5e+06

0 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05

  p  r  e  s  s  u  r  e   (   d  y  n  e   /  c  m   ^   2   )

time (seconds)

"pressure.dat"

FIGURE 7.5

Pressure vs. time applied to base of nozzle for modeling piezo-electric device.

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7.7.1.1 Validation of Contact Angle; Relaxation of Meniscus to Static Shape

In this section we initialize a horizontal meniscus within a 2D axisymmetric cylin-

drical nozzle. The parameters used are similar to the conditions that exist in an ink-jet

nozzle. The initial length of the meniscus is 36 µ. Gas is on top and liquid is onthe bottom. See Figure 7.6 for a diagram of initial conditions. The thick curved line

in Figure 7.6 represents the expected final solution. The meniscus will relax to the

FIGURE 7.6

Initial free surface. Contact-angle boundary condition set at = 45◦. Thick

curved line represents expected static solution. Effective fine grid resolution 128

× 128.

shape that minimizes surface energy. If we assume zero gravity, then the static shape

will be the part of a sphere that intersects the nozzle at the appropriate contact angle.

In Figure 7.7 we compare our computed static shape with the expected shape when

the contact angle is set at = 45◦, the surface tension coefficient is 40 dyne/cm, and

the viscosity of the liquid is 0.05 g/(cms). In Figure 7.8 we plot the kinetic energy

vs. time as the meniscus relaxes to its static shape.As a remark, we have performed the same test above using the 3D version of our

code with similar results; although, for the 3D test, the interfacial thickness for the

free surface has to be set at = 4x instead of  = 3x.

7.7.2 3D Ship Waves

In Figure 7.9, we show a volume rendering of adaptive computations of flow past

a model Navy DDG 5415 ship. In Figure 7.10, we show the x-z slice of the ship flow.

The Froude number for this problem is F 2r = U 2/(gL) = 0.41. We specify periodic

boundary conditions in the x-direction and no-outflow boundary conditions in the y-

direction and at the lower z-direction. The dimensionless length of the ship is 1 unit

and the dimensions of our tank (in terms of dimensionless parameters) is 2×0.5×0.5.

At moderate to high speed, the turbulent flow along the hull of a ship and behind the

stern is characterized by complex physical processes which involve breaking waves,

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FIGURE 7.7

Free surface profile after initial meniscus is allowed to relax to steady state.

Contact-angle boundary condition is set at = 45◦. Thick lines represent

expected static solution. Effective fine grid resolution is 128 × 128.

"energy"

0

100

200

300

400

500

600

700

800

900

  e  n  e  r  g  y

0 5 10 15 20

time (microseconds)

FIGURE 7.8

Kinetic energy vs. time for the relaxation of a meniscus to its final static shape.

Effective fine grid resolution is 128 × 128.

air entrainment, free-surface turbulence, and the formation of spray [22]. Traditional

numerical approaches to these problems, which useboundary-fitted grids, aredifficult

and time consuming to implement. Also, as waves steepen, boundary-fitted grids

will break down unless ad hoc treatments are implemented to prevent the waves

from getting too steep. At the very least, a bridge is required between potential-

flow methods, which model limited physics, and more complex boundary-fitted grid

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methods, which incorporate more physics, albeit with great effort and with limitations

on the wave steepness. Cartesian-grid (embedded boundary) methods are a natural

choice because they allow more complex physics than potential-flow methods and,

unlike boundary-fitted methods, Cartesian-grid methods require minimal effort withno limitation on the wave steepness. Although Cartesian-grid methods are presently

incapable of resolving the hull boundary layer, Cartesian-grid methods can model

wave-breaking, free-surface turbulence, air entrainment, spray-sheet formation, and

complex interactions between the ship hull and the free surface, such as transom-stern

flows and tumblehome bows [49].

FIGURE 7.9

Flow past a model Navy DDG 5415 ship. Effective fine grid resolution is 256 ×64 × 64.

FIGURE 7.10

x-z slice of Flow past a model Navy DDG 5415 ship. Effective fine grid resolution

is 256 × 64 × 64.

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FIGURE 7.11

3D computation of jetting of ink. Solid parts are liquid. ρw/ρa = 816, µw/µa =64. Effective fine grid resolution is 32 × 32 × 256.

References

[1] A.S. Almgren, J.B. Bell, P. Colella, L.H. Howell, and M. Welcome, A con-

servative adaptive projection method for the variable density incompressibleNavier–Stokes equations, J. Comput. Phys., 142, 1–46, 1998.

[2] A.S. Almgren, J.B. Bell, P. Colella, and T. Marthaler, A Cartesian grid pro-

 jection method for the incompressible euler equations in complex geometries,

SIAM J. Sci. Comput., 18(5), 1289–1309, 1997.

[3] J.B. Bell, P. Colella, and H.M. Glaz, A second-order projection method for

the incompressible Navier–Stokes equations, J. Comput. Phys., 85, 257–283,

December 1989.[4] J.B. Bell and D.L. Marcus, A second-order projection method for variable-

density flows, J. Comput. Phys., 101, 334–348, 1992.

[5] J.B. Bell, J.M. Solomon, and W.G. Szymczak, A second-order projection

method for the incompressible navier stokes equations on quadrilateral grids,

in 9th AIAA Computational Fluids Dynamics Conference, Buffalo, June 14–16,

1989.

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Chapter 8

The Solution of Steady PDEs on Adjustable Meshes in Multidimensions Using Local  Descent Methods

M.J. Baines

8.1 Introduction

The method of lines (MOL) is a technique for solving partial differential equations

(PDEs) in which the parameters of a space discretization of the PDE are advanced in

time through the solution of an ordinary differential equation (ODE) system, normally

using a software package. The method can be applied to time-dependent or steady

PDES; in the latter case via convergence in pseudotime. Iterative procedures are in

any case necessary for steady nonlinear PDEs. The MOL has reached a high degree of 

sophistication, as evidenced elsewhere in this volume, and has produced impressive

results.

The purpose of this chapter is to discuss the introduction of mesh movement forsteady PDEs in multidimensions, using mesh locations as additional parameters. In

this way we may seek an optimal mesh at the same time as finding a converged

solution on that mesh. When the mesh locations are included in the parameters of the

space discretization, an extended system of ordinary differential equations (ODEs)

or differential algebraic equations (DAEs) is normally obtained which includes both

mesh and solution parameters in a coupled way. Integration of these equations may

then be carried out using the MOL, although there is a wide variety of approaches.

Adaptation via mesh movement is known as r-refinement. The underlying ideais that the numerical solution of PDEs, particularly those that have rough solutions,

should use all available resources and the mesh is one of these resources. In r-

refinement the solution is adaptively improved by mesh relocation, normally using a

fixedamountof resource. In its simplest formthe numberofnodes remains unchanged

and, provided there is no change in connectivity, there is a fixed data structure.

There are, however, a number of special difficulties with algorithms which involve

mesh movement. First, there is generally no information within the problem about

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how the mesh should be moved and so prescription of the movement is very much in

the hands of the algorithm designer. Although there is sometimes an obvious choice

for the mesh velocity, for example in Lagrangian fluid codes where the mesh is moved

with the velocity of the fluid, there is in general no physically identifiable choice.Second, many PDEs are derived from the application of a physical principle, for

example conservation of mass, in a fixed frame of reference. If the frame of reference

moves, the PDE must be modified and may not retain the physical properties on which

it is based. Third, mesh movement algorithms are essentially nonlinear and exhibit a

high degree of complexity.

Another major difficulty is the possibility of mesh tangling. In one dimension this

simply means node overtaking. In two dimensions, to take an example, a moving

triangulation in which a node of a triangle crosses an opposite side may lead tothe breakdown of a method, either because of singularity at the point of crossing

or because of the inability of the method to function on an invalid triangulation.

Therefore, constraints are often built into a method to avoid tangling.

There are two main techniques for the movement of nodes. The most well-known

technique in this area is that of equidistribution, but we shall be mostly concerned

with techniques that use optimization, since they are valid in multidimensions. The

two techniques overlap in some formulations. Equidistribution is a one-dimensional

concept although there have recently been some significant advances in generalizingthe idea to two dimensions. Links with an approximate form of multidimensional

equidistribution are described in the penultimate section of the chapter. We shall

consider two kinds of functional to be minimized, normally associated with two

different types of PDE. The first is a class of variational principles which generate

PDEs of Euler–Lagrange type, which of their nature are of second order. The second

is the L2 norm of the residual associated with a discretization of the PDE, which can

be used for first-order equations and systems. Both finite-element and finite-volume

discretizations will be discussed.

The existence of a functional allows at least two different approaches to generate

solutions (and meshes). In the more standard approach the full (augmented) ODE

system of normal equations can be solved by the MOL, which we shall refer to as the

global approach. In the other approach we use a descent method on the functional,

which can be implemented in a local manner (node by node) sweeping through the

mesh, which we shall refer to as the local approach.

An early moving-mesh method was the moving finite element (MFE) method [1,

5, 4], which uses piecewise linear finite elements and generates the augmented ODE

system from minimization of the L2 norm of the residual of the PDE over the time

derivatives of both the nodal positions and the solution parameters. The method

is a Galerkin method in which the test functions span the space of the Lagrangian

time derivatives. The method is truly multidimensional and has had some notable

successes, particularly for parabolic problems. However, it also showed up one of 

the difficulties in using piecewise linear approximation in a moving node context,

namely an indeterminacy when the solution and the mesh are simultaneously trying

to represent a linear manifold. This problem occurs when the local curvature of the

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solution manifold vanishes. For this reason (as well as the difficulty of mesh tangling),

regularizing terms were added to the L2 norm in the MFE method with a number of 

adjustable parameters. However, the effectiveness of the method was found to depend

crucially on the manipulation of these parameters and the method has not found favorwith practitioners.

Nevertheless, for a class of steady problems derived from variational principles,

the MFE method gives an optimal solution and the time-dependent form may be used

as an iterative method to drive solutions to steady state (see Section 8.2).

We commencethischapter byconsidering the roleof the MFE method in the context

of optimization. Although originally formulated as an L2 minimization, the method

is not an optimization method in the usual sense but simply an extended weak form

of the PDE. On the other hand, for steady equations of variational type, it has beenshown in [2] that the weak forms correspond to the optimization of a minimization

principle in a discrete space.

The MFE philosophy incorporates a global MOL approach to the solution of the

normal equations. An example of this approach for the steady MFE method is given

in [2]. However, if a functional is available, there is the alternative of sweeping

through the mesh using local descent methods. This is the central theme of this

chapter. Such an approach to optimization using minimization principles is given

in [3] and is described in Section 8.3. A possible finite volume formulation is also

proposed.

Section 8.4 is devoted to least-squares minimization, of particular relevance to

first-order equations and systems. The least squares MFE (LSMFE) method [11]

and a corresponding finite-volume method [7] are described, which use global and

local approaches to the solution procedure, respectively. For the important case of 

conservation laws, the finite-volume procedure may be extended to systems [15], and

a description of this technique forms Section 8.5.

Finally, there is a section on the links with equidistribution (in one dimension) and

approximate equidistribution (in higher dimensions), and a summary section.

8.2 Moving Finite Elements

The moving finite element (MFE) method [1, 5, 4] for the time-dependent PDE

ut  = Lu , (8.1)

where u is a function of x and t , and L is a space operator, is a semi-discrete moving-

mesh, finite-element method in which the node locations are allowed to depend on

time. It is based on two weak forms of the PDE which can be derived from the

minimization of theL2 norm of theresidualover thetime derivatives of theparameters.

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Theapproximate solution U is anexplicit functionof the Xj (t) (the nodal positions)

of the form

U  = j 

U j ψj (x) (8.2)

where U j  are coefficients and the ψj (x) are piecewise linear-basis functions. Using

the result

∂U 

∂Xj 

= (−∇ U ) ψj  (8.3)

(see, e.g., [5]) the derivative of U  with respect to t  becomes

U t  =∂U 

∂t |movingX =

∂U 

∂t |fixedX +

j 

∂U 

∂Xj 

.dXj 

dt 

=d U 

dt  +

j 

(−∇ U ) ψj .dXj 

dt 

=.

U  −∇ U..

X (8.4)

where the independent U  and X functions have time derivatives

.

U =d U 

dt =

j 

dU j 

dt ψj  ,

.

X=dX

dt =

j 

dXj 

dt ψj  (8.5)

which are taken to be continuous functions, corresponding to the evolution of a

continuous piecewise linear approximation.

From (8.1) and (8.4) minimization of the square of the L2 residual U t  − LU 2

L2

over the coefficients.

U j ,.

Xj  then takes the form

min.

U j ,.

Xj 

.

U  −∇ U..

X −LU 

2

L2

(8.6)

and, using (8.5), gives the MFE or extended Galerkin equations

ψj ,.

U  −∇ U..

X −LU  = 0 (8.7)(−∇ U ) ψj ,

.

U  −∇ U..

X −LU 

= 0 (8.8)

Substituting for.

U  and.

X from (8.5) gives a nonlinear system of ODEs for U j  and

Xj  containing an extended MFE mass matrix. The system may be solved globally

for the unknowns U j  and Xj  by a stiff ODE package, as in the MOL.

The basic method has intrinsic singularities, however. If the gradients ∇ U  have

components whose values are equal in adjacent elements (dubbed “parallelism”

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in [1]), the system of equations (8.7)/(8.8) becomes singular and must be regularized

in some way. If the area of an element vanishes (i.e., a triangle becomes degener-

ate) the system again becomes singular and special action is required. In Miller’s

MFE method, penalty functions are added to the L2 norm of the residual in (8.6)(see [1, 4, 5]).

Although a full understanding of the MFE method is incomplete, in the steady limit

the resulting mesh has a significant optimal property. We now consider this limit.

8.2.1 MFE in the Steady-State Limit

In many cases the MFE method may be used to generate weak forms for the

approximate solution of the steady PDE

Lu = 0 (8.9)

by driving the MFE solutions to convergence in pseudotime, although not always.

For scalar first-order PDEs, the MFE method is known to move the nodes with char-

acteristic speeds [5] which do not generally settle down to a steady state.

From (8.6) the MFE method in the steady case implements the minimization

min.

U j ,.

Xj 

LU 2L2

(8.10)

and the steady-state solution satisfies the weak forms1

−∇ U 

ψj , LU 

= 0 . (8.11)

Although.

U  and.

X no longer appear in LU , the minimization is over the span of 

their time derivatives ] which is the space spanned by the functions {ψj , (−∇ U ) ψj }.

In order to describe the optimal property of the steady MFE method, we recall the

origin of PDEs of Euler–Lagrange type.

8.2.2 Minimization Principles and Weak Forms

A standard result in classical analysis is that minimization of the functional

J ( F) =

 F(u, ∇ u)dx (8.12)

over a suitable class of functions yields the PDE

Lu = − ∂F ∂u

+ ∇ . ∂F ∂∇ u

= 0 . (8.13)

By a similar argument, minimization of the functional (8.12) over the finite dimen-

sional space spanned by the {ψj (x)} [i.e., over approximations of the form (8.2)] on

a fixed mesh yields the weak formψj ,

∂F 

∂U 

+

∇ ψj ,

∂F 

∂∇ U 

= 0 . (8.14)

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8.2.3 An Optimal Property of the Steady MFE Equations

It has been shown in [2] that minimization of the functional (8.12) over functions

in the MFE approximation space, spanned by ψj (x),−∇ 

U )ψj (x), yields the steadyMFE equations (8.11). Hence, the steady MFE equations provide an optimal U  and

X for variations of the functional (8.12) within this space.

These weak forms are

∂

∂U j 

 F(U, ∇ U)dx =

ψj ,

∂F 

∂U 

+

∇ ψj ,

∂F 

∂∇ U 

= 0 (8.15)

as in (8.14) and

∂∂Xj 

 F(U, ∇ U)dx = ψj , ∂F ∂x

+ ∇ ψj ,F  − ∇ U  · ∂F ∂∇ U 

= 0 (8.16)

where the identity

∇ ·

F  − ∇ U  ·

∂F 

∂∇ U 

=

∂F 

∂x+

∂F 

∂U ∇ U  −

∇ ·

∂F 

∂∇ U 

∇ U  (8.17)

has been used to derive a form of (8.16) which is formally suitable for piecewise

linear approximation. In carrying out the integration by parts to arrive at (8.16) we

have used the fact that the continuous piecewise linear finite element basis functionψj  vanishes on the boundary of the patch. The result in [2] is that (8.15) and (8.16)

are identical to the weak forms (8.11).

It may be possible to use the time-dependent MFE method as an iterative procedure

to generate locally optimal meshes in the steady state. This approach has been used

in [2] to generate optimal solutions with variable nodes to a number of examples of 

PDEs of Euler–Lagrange type. A partially regularized form of the MFE method is

used in order to avoid singular behavior and a global MOL solver employed to extract

the solution. For further details see [2]. However, since the iteration need not be timeaccurate, the MFE mass matrix may, if desired, be replaced by any positive definite

matrix.

We now come to the central theme of this chapter, which is to consider the role

of descent methods in generating solutions of problems of this type using a local

approach.

8.3 A Local Approach to Variational Principles

Since minimization principles provide a functional to monitor and reduce, it is

possible to take advantage of standard optimization procedures in generating local

minima. For example, procedures based on descent methods give the freedom to use

a local approach to iteration, which significantly reduces the complexity of problems

involving mesh movement. First we recall the nature of descent methods.

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8.3.1 Descent Methods

Descent methods are based upon the property that the first variation of a functional

J with respect to a vector variable

Y ,

δJ  =∂J 

∂Y δY  = gT δY  (8.18)

say, is negative when

δY  = −τ g = −τ ∂J 

∂Y (8.19)

for a sufficiently small positive relaxation parameter τ , and therefore reduces J .

Choice of τ  is normally governed by a line search or a local quadratic model.

The left-hand side of (8.19) may be preconditioned by any positive definite matrix.

The Hessian gives the full Newton approach but may be approximated in various

ways.

In the present context of r-adaptivity, a local approach is advantageous which

consists of updating the unknowns one node at a time (scalar Y ), using only local

information. Moreover, U j  and Xj  may be updated sequentially, which permits close

control of the mesh movement. The updates may be carried out in a block (Jacobi

iteration) or sequential (Gauss–Seidel) manner. Descent methods of this type have

been used by Tourigny and Baines [6] and Tourigny and Hulsemann [3] in the L2

case and by Roe [7] and Baines and Leary [8] in the discrete case.

First we mention a local approach to L2 best fits with adjustable nodes.

8.3.2 A Local Approach to Best Fits

A minimization based on a local approach was used in [9] to generate algorithmsto determine best discontinuous piecewise constant and piecewise linear L2 fits

to a given function in one or two dimensions, with adjustable nodes. The conver-

gence of the one-dimensional algorithm was subsequently investigated in [6] and

the method shown to reduce the L2 norm of the residual error monotonically. The

two-dimensional algorithm was modified in [6] and a procedure for improving the

connectivity introduced. Convergence of the method was also considered in [6] for

successively (globally) refined meshes.

A special feature of these algorithms is their local nature, the nodal and solutionupdates being carried out one node at a time within sweeps through the mesh. This

approach not only reduces the complexity of the problem but also allows for mesh

tangling to be avoided relatively easily using a limiter (see, e.g., [19]). Moreover,

owing to the existence of a functional to minimize, edge swapping and node removal

are readily incorporated.

We now turn to the approximation of PDEs and consider minimization principles

using the local approach.

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8.3.3 Direct Optimization Using Minimization Principles

An early attempt to include mesh adaptation into a minimization principle was

due to Delfour et al. [10], who sought a finite-element solution with free nodes fora variational formulation of an elliptic PDE but found significant problems with the

complexity of the equations and with mesh tangling.

More recently, an iterativealgorithmwith variablenodes for thefinite-elementsolu-

tion of minimization problems has been described in [3] using the localapproach. The

criterion is that the mesh should be such that a variational “energy functional” evalu-

ated at the finite-element approximation is reduced. Each node is treated separately

in sweeping through the mesh. The nodal positions are updated by a steepest descent

procedure, during which a sequence of local finite-element problems is solved, each

involving very few degrees of freedom. The order of sweeping through the meshis based on the size of the local residuals. The method is applied to a variety of 

minimization principles in two dimensions.

We describe the essence of the method here. A finite-element approximation U  is

sought to optimize a convex energy functional of the form (8.12) in a subspace V hsuch that

J(U) = minV ∈V h()

J ( V ) (8.20)

where V h is the set of piecewise linear functions defined on a triangulation which

is allowed to deform. The minimization is conceived in terms of solving a sequence

of local problems on patches of triangles {T j } surrounding node j  (see Figure 8.1)

and sweeping through the mesh. Each local approximation U  is computed using the

FIGURE 8.1

A local patch of elements surrounding node j .

normal equations {T j }

∂F 

∂U (U , ∇ U )ψj  +

∂F 

∂∇ U (U , ∇ U ) · ∇ ψj 

dx = 0 (8.21)

∀ψj  ∈ V h [cf. (8.14)]. In a local patch such as that in Figure 8.1 the computation of 

U  at node j  on a fixed mesh involves the determination of only a single unknown,

which can be carried out cheaply using only local information on the patch.

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New local nodal positions X = Xnewj  are sought within the iteration, with the

corresponding solution U  = U new, such that J(U) is reduced. A steepest-descent

method is used. Thus, if Xj   is an interior node, a new mesh location is sought along

the line given by

Xnewj  = Xj  − τ 

∂J 

∂Xj 

, (8.22)

where the relaxation factor τ  is chosen by a line search. Although this requires the

solution of a sequence of finite-element problems of the form (8.21), these are small,

local problems.

The sequence of nodal updates is carried out in a Gauss-Seidel manner. Edge

swapping is interleaved with the grid-movement algorithm, an edge being swapped if it leads to a lower energy. Global mesh refinement is also included in the algorithm,

refinement taking place after the algorithm for the current number of nodes had

converged. That is, starting from a coarse mesh the algorithm is used to optimize the

mesh; this optimal mesh is then uniformly refined to provide the starting mesh for the

next refinement level. The possible occurrence of degenerate triangles is overcome

by the use of a node-deletion algorithm. Convergence to the “global” solution relies

on the sweeps through the mesh.

Convergence rates in a test problem on Laplace’s equation involving a re-entrant

corner showed that, whereas convergence on a uniform mesh was sub-optimal, theapproximation on theadapted meshesconstructed in this way converged at theoptimal

rate. Figure 8.2 shows two of the meshes obtained using the algorithm with successive

globally refined meshes. There is an exact solution of this problem of the form

r2/7 sin( 27

)θ  (for full details see [3]).

FIGURE 8.2

Two meshes for the re-entrant corner problem.

8.3.4 A Discrete Variational Principle

A discrete form of (8.12) (obtained by quadrature) is

J d (F ) =

T 

S T F(U, ∇ U )T  (8.23)

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where the suffix T  runs over all the triangles of the mesh, S T   is the area of triangle

T , and the overbar denotes the average value of the argument over the vertices of the

triangle T .

Differentiation of (8.23) with respect to U j   gives (see [22])

∂J d 

∂U j 

=

T J 

1

3S T 

∂F 

∂U 

j 

+∂F 

∂ (∇ U )T 

· nj 

(8.24)

where nj   is the inward normal to the side opposite node j  scaled by the length of that

side. Setting this gradient to zero gives the finite volume weak form corresponding

to the finite-element weak form (8.14).Differentiation with respect to Xj   gives (see [17])

∂I d 

∂Xj 

=

T j 

1

3S T 

∂F 

∂x

T 

+

−

∂F 

∂U y,

∂F 

∂U x

j 

U j 

+

T j 

∂F 

∂∇ U .∇ U  − F 

T 

nj . (8.25)

Setting this gradient to zero gives the companion weak form to (8.24), corresponding

to the second finite-element weak form (8.16).

We may approximate (∇ U )T   by the gradient of the linear interpolation between

the corner values of U  in the triangle T , given by (see [13])

(∇ U )T  = −

U Y 

XY 

,

U X

−

Y X =

Y U 

XY 

,−

XU 

Y X (8.26)

where the sums run over the vertices of the triangle T  and X, Y, U  denote

the increments in the values of X, Y, U  taken counterclockwise across the side of T 

opposite the corner concerned (see Figure 8.1). This is the same as the piecewise

linear approximation used in the finite-element case. In the same notation as above,

the area of the triangle T  is

S T  = 12

XY  = − 12

Y X . (8.27)

The expressions (8.24) and (8.25) with ∇ U  given by (8.26) provide gradients for

a steepest-descent method for minimizing (8.23). Examples of this approach will be

given in the next section.

We turn now to least-squares methods, which extend the applicability of the tech-

niques already described to first-order equations and systems.

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8.4 Least-Squares Methods

The steady MFE and local minimization techniques are valid and useful if there

exists a minimization principle for the PDE, but are not available in other cases where

no such principle exists, in particular for first-order equations and systems. (Euler–

Lagrange equations, by their nature, are of at least second order.) However, the same

minimization techniques can also be applied to least-squares methods, where the

“energy functional” is the square of the norm of the residual. We describe two such

methods, one of MFE type exploiting the optimal property of the steady MFE method

described in Section 8.2.3 and the other arising from a finite-volume approach, similar

to that in Section 8.3.4.In this section, we shall assume that L is a first-order space operator, depending on

x, u, and ∇ u only.

8.4.1 Least-Squares Moving Finite Elements

Although the L2 norm (8.6) was minimized in formulating the MFE method de-

scribed in Section 8.2, this minimization is only carried out over the velocities.

U j  and.

Xj  and is thus not a true minimization at the fully discrete level. Variations in U j  andXj  are treated as independent of those in

.

U j  and.

Xj  and are ignored. It can therefore

be seen from (8.6) that the method is simply a linear least-squares problem, generat-

ing the weak forms (8.7) and (8.8). It is more useful to say that the minimization is

carried out in the space spanned by the basis functions {ψj , (−∇ U )ψj }.

By contrast, a full minimization of the L2 norm in (8.10) may be carried out in

the steady case over the nodal coordinates Xj  and the coefficients U j . This is the

approach of the recent least squares moving finite element (LSMFE) method [11].

This is a nonlinear least-squares problem, so only a local minimum can be expected.Consider then the minimization of (8.10) over these parameters, which leads to the

two weak formsLU,

∂

∂U j 

LU 

=

LU, (−∇ U )

∂

∂U j 

(LU )

+

 1

2(LU )2 ψj n ds = 0

(8.28)

(

n is the unit normal) which may be written

∂ (LU )2

∂U , ψj 

+∂ (LU )2

∂∇ U , ∇ ψj 

= 0 (8.29)

and ∂ (LU )2

∂x, ψj 

+

(LU )2 − (∇ U ) .

∂ (LU )2

∂∇ U 

, ∇ ψj 

= 0 , (8.30)

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where the identity (8.17) has been used with F  = 12 (LU)2.

Referring back to (8.15) and (8.16) we see that Equations (8.29) and (8.30) are the

steady MFE equations for the PDE

0 = ut  = −∂ (Lu)2

∂u+ ∇ ·

∂ (Lu)2

∂∇ u (8.31)

which corresponds to the Euler–Lagrange equation for the minimization of the least

squares functional Lu2L2

. For the LSMFE method the optimal property holds just

as for the variational method in Section 8.3.

It is natural to solve the nonlinear system of Equations (8.29) and (8.30) using the

MFE time-stepping method, but any other convenient iteration can be used. In [11]

the mass matrix of the MFE method is replaced by a Laplacian-regularization matrix.

8.4.2 Properties of the LSMFE Method

The LSMFE method has the following properties:

• The weak forms (8.29) and (8.30) arising from these variations correspond to

Equations (8.15) and (8.16) with F  given by 12 (LU )2 and, therefore, have the

optimal property.

• In the LSMFE tests carried out in [11] on scalar first-order steady equations the

nodes no longer move with characteristic speeds but instead move to regions

of high curvature. This is to be expected because the least-squares procedure

embeds the original first-order equation in the second-order  equation (8.31),

and it is already known that, for Laplace’s equation in one dimension, the final

positions of the nodes in the MFE steady limit asymptotically equidistribute a25

power of the second derivative of |u| [12].

• A third property is only stated here. In the particular case where LU  takesthe form of a divergence of a continuous function, a modification of the result

discussed in Section 8.6.3 below shows that, asymptotically, minimization of 

LU 2L2

is equivalent to an equidistribution of LU  over each element in the

particular sense described there. For example, in the case where

Lu = ∇ · (au) (8.32)

with constant a, and u is approximated by the continuous piecewise linear

function U , the LSMFE method asymptotically equidistributes the piecewiseconstant residual LU  = ∇ · (aU) in each element. Within a coupled iteration

scheme this ensures that convergence proceeds in a relatively uniform manner.

An illustration of the results of the method for a 2D circular advection problem is

given in Figure 8.4, the mesh being very similar to that given by the least-squares,

finite-volume method described there.

We now consider minimization of least squares functionals with moving nodes in

which the approximations are of finite-volume type.

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The fluctuation is defined as the integral of the residual in a triangle T 

φT  =  T 

LUdx , (8.37)

or in its finite-volume form, using quadrature,

φT  = S T LU . (8.38)

The discrete least-squares norm of the residual, from (8.33) and (8.35), is therefore

given by

LU 2

d = T 

φ2T 

S T 

=1

3 j  T j 

φ2T 

S T 

. (8.39)

With thealternativeweight S 2T  in (8.36), the discrete least-squares norm of the residual

is simply the l2 norm of the fluctuation,

|LU |2d  = φ2

l2=

T 

φ2T  =

1

3

j 

T j 

φ2T  . (8.40)

By including mesh variables in the least squares minimization of (8.39) Roe in [7]alleviated the counting problems with the use of a fixed grid where, even though

the norm of the residuals over a patch may vanish, the element residuals do not,

leading to an unsatisfactory solution (see [8]). When nodal positions are included

in the minimization process, the number of degrees of freedom is increased and at

convergence the element fluctuations are driven close to zero and a much improved

solution is obtained.

A steepest descent methodwas used in [7] in which local updates of the solutionand

the mesh were made with a safe value of the relaxation parameter τ . The convergence

of the algorithm is extremely slow but can be improved by using a more sophisticated

line search [14]. However, what enormously improves the convergence rate is an

updating mechanism which does not come from a full least squares descent method

but which takes updates only from the upwind direction [8].

As an illustration consider the scalar two-dimensional advection equation

a(x).∇ u = 0 . (8.41)

Then the fluctuation may be written

φe = −1

2

3ei=1

(aei U ei Y ei − bei U ei Xei ) (8.42)

where a = (a,b) = (a(Xei , Y ei ), b(Xei , Y ei )).

The steady-state residual is

a(x).∇ U  (8.43)

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and, from (8.38),

φT  =(a.∇ U )2

S T 

. (8.44)

Then the derivatives of (8.39) with respect to U j   and Xj   reduce toa.∇ U ,a .

∂ (∇ U )

∂U j 

d 

=

a.∇ U,

∂

a.∇ U 

∂Xj 

d 

+1

2

{T j }

a.∇ U T 

2 ∂S T 

∂Xj 

= 0 ,

(8.45)

subject to boundary conditions, where from (8.26) and (8.27)

∂ (∇ U )T j 

∂U j 

=1

2S T j 

nj   (8.46)

∂

a.∇ U 

T j 

∂Xj 

= U j 

−b

a

−

1

2

S T j 

(∇ U )T j . (8.47)

Recall that nj   is the inward normal to the side of the triangle opposite node j  scaled

by the length of that side and U j   is the increment in U  across that side, taken

counterclockwise (see Figure 8.1).Equation (8.45) may therefore be written as:

T j 

a.∇ U 

T 

a.n

T 

= 0 (8.48)

and

T j  a.∇ U T U j 

−b

a −1

2 a.∇ U 2

T n = 0 . (8.49)

We observe that (8.48) is identical to (8.29) when LU  = a.∇ U , noting that ∇ U  is

constant and ∇ φ = S −1T  n. However, (8.49) does not reduce to (8.30) even when a is

constant.

We now show an example taken from [7].

8.4.5 Example

Let a(x) = (y, −x) in a rectangle −1 ≤ x ≤ 1, 0 ≤ y ≤ 1. Then the solution of 

(8.41) is a semicircular annulus swept out by the initial data, here chosen to be

U  =

1

0

−0.6 ≤ x ≤ −0.5

otherwise .(8.50)

Results are shown in Figures 8.3 and 8.4 for a fixed and moving mesh, respectively,

taken from [15].

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FIGURE 8.3

Initial mesh and solution for the circular advection problem.

FIGURE 8.4

Final mesh and solution for the circular advection problem.

As expected, the solution on a fixed mesh is poor. However, when the mesh

takes part in the minimization the norm is driven down to machine accuracy. The

redistribution effected by the least squares minimization forces global conservation

and equidistributes φ among the triangles (see Section 8.6.3) leading to more uniform

convergence. Cell edges have approximately aligned with characteristics in regions

of non-zero φ, allowing a highly accurate solution to be obtained. Essentially the

same final mesh is obtained by the LSMFE method of Section 8.3.

The left-hand graph in Figure 8.5 shows the convergence of the solution updating

procedure on the fixed mesh using (a) steepest descent with a global relaxation factorτ  = 0.5, (b) optimal local updates using a quadratic model, (c) optimal local updates

over downwind cells only. Convergence is much improved in (b) and (c). Even

though (c) is not monotonic it converges very quickly, albeit to a higher value of the

functional, due to the nature of the procedure.

The convergence rates obtained when the nodes are allowed to move are shown

in the right-hand graph in Figure 8.5. The iteration is started from the converged

solution on the fixed mesh and uses (a) steepest descent with the global relaxation

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FIGURE 8.5

Comparison of convergence histories.

factors τ  = 0.5 for the solution and τ  = 0.01 for the meshpoints, (b) a line search

using a Newton iteration, (c) a line search with updates over downwind cells only.

A small amount of mesh smoothing was included in (b) and (c). In particular,

(b) became stuck in a local minimum if more iterations are used. Node locking was

a problem with the full least squares approach. Node removal or steepest descent

updates may be used to alleviate this problem ([6, 3]) but when tried in [14] still took 

over 1000 iterations and so were not competitive when compared to the upwinding

approach, which yielded the best result.

Discrete least squares solutions of the Stokes Problem have been considered in

[21]. Here the two different discrete norms (8.39) and (8.40) were compared, one

with the area weighting and one without, but little difference was seen in the results.

8.5 Conservation Laws by Least Squares

Finally we describe an advance in locating shocks in the solution of systems of 

nonlinear hyperbolic equations. In [15] the least squares minimization technique was

used for systems, combining shock capturing techniques with shock fitting.

A general system of conservation laws is of the form

divf (u) = 0 = A(u).∇ u (8.51)

where A is a vector of the Jacobian matrices (A,B)T . The integral form is 

f (u).nd = 0 (8.52)

where n is the inward facing unit normal.

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It is assumed that f  is approximated by a piecewise linear function F. Then the

fluctuation in triangle T  is defined to be

T  = −  T 

divFdx =  ∂T 

F.n ds (8.53)

where ∂T  is the perimeter of T . The average residual is also defined as

RT  =1

S T 

 ∂T 

F.n ds =T 

S T 

(8.54)

where S T   is the area of triangle T .

Since F is assumed to be linear in the triangle a trapezium rule quadrature can be

used to write the fluctuation in triangle e, from (8.53), as

e = 1

2

Fe1 + Fe2

.ne3 +

Fe2 + Fe3

.ne1 +

Fe3 + Fe1

.ne2

, (8.55)

where nei (i = 1, 2, 3) is the inward unit normal to the ith edge of triangle e (opposite

the vertex ei), as shown in Figure 8.6, multiplied by the length of that edge.

FIGURE 8.6

A general triangular cell e.

It is easy to verify that, for any triangle,

ne1 + ne2 + ne3 = 0 , (8.56)

so the fluctuation (8.55) may be written as

e = −1

2

Fe1.ne1 + Fe2.ne2 + Fe3.ne3

(8.57)

or, since nei = (Y ei , −Xei ), as

e = −1

2

3ei=1

(Fei Y ei − Gei Xei ) (8.58)

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[cf. (8.42), where F = (F, G) and (e1X, e1Y ) = (Xe2 − Xe3, Y e2 − Y e3) denotes

the difference in X taken across the side opposite node e1 in a counterclockwise sense

(with similar definitions for (e2X, e2Y ) and (e3X, e3Y )] (see Figure 8.6). A

useful dual form of the fluctuation is obtained by rewriting (8.58) as

e = 1

2

3ei=1

(Y ei Fei − Xei Gei ) . (8.59)

We aim to set the fluctuationse to zero in order to minimize a vector form of (8.35).

Two special systems of interest are the Shallow Water equation system and the

Euler equations of gasdynamics. Details of these systems are given in [14].

8.5.1 Use of Degenerate Triangles

In the presence of shocks least squares methods give inaccurate solutions which are

unacceptable. In [15] a way of combating this problem is shown which is to divide

the region into two domains and introduce degenerate triangles at the interface. The

least squares method with moving nodes is then used to adjust the position of the

discontinuity, as in shock fitting methods.

An initial approximate solution to the equations can be found by any standard

method. In [15], a multidimensional upwinding shock capturing scheme is used. An

initial discontinuous solution is then constructed by introducing degenerate (vertical)

triangles in the regions identified as shocks, using a shock identification technique.

This step is carried out manually in [15] although degenerate triangles can be added

automatically using techniques that exist in the shock fitting literature (see, for exam-

ple, [16]). The corners of the degenerate triangles are designated as shocked nodes

and these form an internal boundary, on either side of which the least squares method

is applied in the two smooth regions where it is known to perform well. The posi-

tion of the discontinuity is then improved by minimizing a shock functional which is

derived from (8.40).Consider the interface shown in  Figure 8.7. The fluctuations d 1 and d 2 in

adjacent degenerate triangles d 1 and d 2 on the edges i = (iL, j L) and j  = (iR, j R)

are, from (8.57),

d 1 = −1

2

Fi

.niL

d 2 = −1

2

Fj 

.nj R

(8.60)

respectively, where the square bracket denotes the jump across the discontinuity.

Then

T d 1d 1 +T 

d 2d 2 =

1

4

[Fi ].niL

T  [Fi ].niL

+

[Fj ].nj L

T  [Fj ].nj L

(8.61)

and the functional T ∈D

T T T  (8.62)

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FIGURE 8.7

Degenerate triangles d 1, d 2.

[cf. (8.40)] is minimized to improve the position of the shock, where D is the

set of degenerate triangles. This norm is always bounded, even at shocks where U

is discontinuous. On the other hand, the average residual, given by (8.39), is not

bounded at shocks.

A recent result of Nishikawa et al. [28] somewhat surprisingly suggests that the

capability of fluctuation splittingmethods to capture characteristics or shocksdepends

on the quadrature used in defining the fluctuation.A descent least squares method is used on (8.62) to move the shocked nodes into

a more accurate position. The procedure is interleaved with a descent least squares

method on (8.39) for the smooth solution on either side.

When updating thenodal positionsXiLand XiR

it is required that theyhave the same

update (so that the cell remains degenerate). The update comes from minimization

with respect to their common position vector. Degenerate quadrilaterals can be used

instead of degenerate triangles.

We reproduce two results using this technique, taken from [15].

8.5.2 Numerical Results for Discontinuous Solutions

Results are shown from two problems which exhibit discontinuous solutions, one

for the Shallow Water equations and the other for the Euler equations of gasdynamics.

The Shallow Water equations system can be used to describe the problem of a tran-

scritical constricted channel flow which exhibits a hydraulic jump in the constriction.

The computational domain represents a channel of length 3 meters and width 1 meter

with a 10% bump in the middle third. The freestream Froude number is defined to be

F ∞ = 0.55, the freestream depth is h∞ = 1m [and the freestream velocity is given

by (u∞, v∞) = (1.72, 0)]. An initial solution is found by the Elliptic-Hyperbolic

Lax–Wendroff multidimensional upwinding scheme of Mesaros and Roe, see [17].

The hydraulic jump is then located and degenerate quadrilaterals added at the approx-

imate position of the shock. The best position of the shock is then sought using a

least squares descent method with degenerate triangles, moving the nodes to improve

the position of the shock.

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FIGURE 8.8

Mesh and height contours for the Shallow Water example.

Results are shown in Figure 8.8 which shows the meshes and solution depth con-

tours obtained. A bow-shaped hydraulic jump which is strongest at the boundariesis predicted. This agrees with the solution obtained using a shock capturing solution

on a very fine mesh. Here it is achieved sharply at very much less cost.

In the second example the Euler equations of gasdynamics are considered written

in conserved variables.

The example chosen exhibits the shock fitting capabilities of the method for a purely

supersonic flow which has an exact solution [18]. The computational domain is of 

length 3 meters and width 1 meter. Supersonic inflow boundary conditions, given by

U (0, y) = (1.0, 2.9, 0, 5.99073)t 

U(x, 1) = (1.69997, 4.45280, −0.86073, 9.87007)t  , (8.63)

areimposed on theleft andupperboundaries, respectively. At theright-hand boundary

supersonic outflow conditions are applied, while the lower boundary is treated as a

solid wall.

The boundary conditions are chosen so that the shock enters the top left-hand corner

of the region at an angle of 29

◦ 

to the horizontal and is reflected by a flat plate onthe lower boundary. The flow in regions away from shocks is constant. The same

strategy is employed as in the previous example, with the results shown in Figure 8.9

where the mesh and the density are shown. The solution has a shock which comes in

from the top left hand at an angle of 29.2◦ to the horizontal and is virtually constant

apart from the discontinuities, in close agreement with the analytic solution.

The final section in this chapter highlights the links between the minimization

procedures discussed previously and the ideas of equidistribution.

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FIGURE 8.9

Mesh and density for the Euler equations example.

8.6 Links with Equidistribution

The well-known equidistribution principle (EP) in one dimension involves locating

meshpoints such that some measure of a function is equalized over each subinterval

[22]. In one dimension, denoting by x and ξ  the physical and computational coordi-

nates, respectively, define a coordinate transformation

x = x(ξ ) ξ  ∈ [0, 1] (8.64)

with fixed end points x(0) = a , x (1) = b, say. The computational coordinates are

given by

ξ i

=i

N , i = 0, 1,...,N  (8.65)

where N  is the number of mesh points.

A positive monitor function M(u) is chosen that provides some desired measure

of the solution u to be equidistributed. The integral form of the EP is then given by x(ξ)

0

M(u)dx = ξ θ  (8.66)

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where θ  = 1

0M(u)dx. Differentiating (8.66) twice with respect to ξ  gives the

alternative differential form

∂

∂ξ  M(u)∂x

∂ξ  = 0 . (8.67)

In practice a discretized form of (8.67)

M j −1/2

xj  − xj −1

= M j +1/2

xj +1 − xj 

(8.68)

may be solved subject to the boundary conditions x(0) = a ,x (1) = b.

Themethodhasbecomevery popular in many contexts. However, differentmonitor

functions are often required for different purposes [22].Since the monitor function depends on u, which depends in turn on x, an iteration

procedure is needed to solve (8.68). More specifically, the monitor function depends

on the solution of the PDE, so Equation (8.68) should be thought of as just one step

in an iterative algorithm for both the mesh and the solution.

In iteratively solving (8.68) a single step of an iteration for each equation may be

generated and the mesh iterations interleaved with the iterations for solving the PDE.

These iteration steps may be chosen to involve only one node at a time (so that the

iteration is tantamount to a sweep through the mesh) and then we have a sequence of 

local problems as in Sections 8.3 through 8.5 above.

8.6.1 Approximate Multidimensional Equidistribution

Equidistribution was conceived as a technique for approximation in one dimension.

Nevertheless, recently there have been important developments in multidimensional

equidistribution, (see [23, 20]). However, we shall discuss only approximate gener-

alizations to higher dimensions here, since these are simple to implement and relate

to other ideas in this chapter.A formal generalization of (8.68) ise∈{T j }

M e

xn

e − xnj 

= 0 (8.69)

where xne  is the centroid of triangle e, M e is a weight and {T j } is the set of triangles

surrounding node j . (see Figure 8.1). Although the formula is convex, there are

examples of meshes in which mesh tangling can take place in this case [19].

The formula (8.69) is not a statement of equidistribution, but it does have an inter-polatory status, intuitively equidistributing in the two limits

(a) if M e = constant ∀e, xj  is the average of the centroids of the surrounding

triangles,

(b) if one M e dominates, say M E , then the nodes cluster toward the centroid of the

element E.

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The positions of the mesh vertices may also be interpreted as the solution of the

least squares minimization problem (see [24, 25])

minxj 

e

M e xj  − xej 2

, (8.70)

8.6.2 A Local Approach to Approximate Equidistribution

Again, the use of iterative techniques facilitates a local approach in which the

iteration may be carried out one node at a time, sweeping through the mesh. Consider

the iteration in which mesh points are moved to weighted averages of the positions

of centroids of adjacent cells.

In one dimension an iteration for the solution of (8.68) is of the form

xn+1j  =

M j − 12

xn

j  + xnj −1

+ M j + 1

2

xn

j  + xnj +1

2

M j − 1

2+ M 

j + 12

. (8.71)

which is convex, convergent, and does not allow mesh tangling [19]. In two dimen-

sions a corresponding iteration for (8.69) is

xn+1j  = e∈{T j } M exne

e∈{T j }M e

. (8.72)

8.6.3 Approximate Equidistribution and Conservation

A link between discrete least squares and equidistribution is described in [20]

where it is shown that least squares minimization of the residual of the divergence of 

a vector field is equivalent to that of a least squares measure of “equidistribution” of 

the residual.The conservation law (8.51) is considered where u is approximated by the contin-

uous approximation U. The fluctuation e is defined as in (8.53) and the average

residual as in (8.54). Then the following identity holds.N 

i=1

S i

N 

i=1

RT T  S T RT 

= N 

i=1

i

2

+1

2

N i=1

N j =1

RT i − RT j 

T S T i S T j 

RT i − RT j 

. (8.73)

Now

N i=1

S T i = (8.74)

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is equal to the total areaof the union of the triangles and may be taken tobe constant.

Moreover, by definition,

N i=1

i = −

N i=1

 e

divF(u)dx =  ∂

F u .ds (8.75)

by internal cancellation, which is independent of interior values of  F and interior

mesh locations.

We may then write (8.73) as

R2

l2=  ∂

F.ds2

+ R2

eq(8.76)

where

R2

l2=

N i=1

RT T  S T RT  (8.77)

corresponding to (8.39) and where

R2eq

= 12

N i=1

N j =1

RT i − Rj T T 

S T i S T j RT i − RT j 

(8.78)

which is a measure of equidistribution of the average residual RT .

A similar result can be derived for the norm (8.40). The identity

N 2l2

≡

 ∂

F.ds

2

+ ||2eq (8.79)

holds, where N  is the number of cells and

2l2

=

N i=1

T T iT i (8.80)

proportional to (8.40), and

||2eq =1

2

N i=1

N j =1

T i −T j T  T i −T j  . (8.81)

If we allow only interior mesh points to be varied, then (8.75) is a fixed quantity

and in any minimization procedure the two norms (8.77) and (8.78) [or (8.80] and

(8.81 ) ] will be minimized simultaneously. The minimization of (8.77) [or (8.80)]

[corresponding to finding a least squares approximation to the solution of (8.51)] is

equivalent to minimizing a measure of equidistribution over the triangles in the sense

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of (8.78) or (8.81). This result holds in any number of dimensions and in an iterative

context encourages convergence to take place in a uniform way.

Finite volume methods of the type discussed here may not give very accurate

solutions. However, as far as the mesh is concerned, high accuracy is not crucial. Afinite volume approach may therefore be sufficiently accurate for the mesh locations

but for a higher order solution a more sophisticated method, such as high order finite

elements or multidimensional upwinding ([26, 27]), may be required for the solution

on the optimal mesh.

8.7 Summary

The MFE method is a Galerkin method extended to include node movement. For

the PDE

Lu = −∂F 

∂u+ ∇ .

∂F 

∂∇ u= 0 (8.82)

the steady MFE equations provide a local optimum for the variational problem

minU j ,Xj 

 F(U, ∇ U)dx (8.83)

in a piecewise linear approximation space with moving nodes.

Solutions of such PDEs may also be obtained by direct minimization of (8.83). A

local approach is possible which is advantageous in reducing the complexity of the

mesh location procedure and in applying constraints which preserve the integrity of 

the mesh. An approach of this kind was described in Section 8.3.

The Least Squares Moving Finite Element method (LSMFE) is a least squares

method for steady first order PDEs which includes node movement. In the steady

state the LSMFE equations for Lu = 0 are equivalent to the steady MFE weak forms

for the PDE

−∂ (Lu)2

∂u+ ∇ .

∂ (Lu)2

∂∇ u

= 0 (8.84)

and therefore provide a local minimum for the variational problem

minU j ,Xj 

 (LU )2 dx (8.85)

Moreover, if LU  is the divergence of a continuous flux function then the flux across

element boundaries is asymptotically equidistributed over the elements.

A least squares finite volume fluctuation distribution scheme with mesh movement,

givenbyRoe in [7] is anadaptive meshmethodbased onminimizationofa weighted l2

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norm of the residual of a steady first order PDE over the solution and the mesh. It also

uses a local approach and a steepest descent algorithm. It lacks the optimal property

of LSMFE but has the property that, if LU  is the divergence of a continuous flux

function, then the flux across element boundaries is equidistributed over the elementsin the sense of (8.78) or (8.81), thus proceeding to the steady limit in a uniform way.

For scalar problems convergence can be greatly accelerated by carrying out the

iterations in an upwind manner.

For problems with discontinuities, the mesh movement technique enables improve-

ment of the location of the discontinuity in a manner akin to shock fitting. By mini-

mizing a measure of the fluctuation in degenerate triangles, an initially approximate

position of the shock can be maneuvered into an accurate position. The descent least

squares method may be used on either side of the shock to gain good approximations

in the smooth regions of the flow.

References

[1] K. Miller, Moving finite elements I (with R.N.Miller) and II, SIAM J. Num.

 Anal., 18, 1019–1057, (1981).

[2] P.K. Jimack, Local Minimization of Errors and Residuals Using the Moving

Finite Element Method, University of Leeds Report 98.17, School of Computer

Science, (1998).

[3] Y. Tourigny and F. Hulsemann, A new moving mesh algorithm for the finite

element solution of variational problems, SIAM J. Num. Anal., 34, 1416–1438,

(1998).

[4] N.N. Carlson and K. Miller, Design and application of a gradient weighted

moving finite element method I: in one dimension. II: in two dimensions, SIAM 

 J. Sci. Comp., 19, 728–798, (1998).

[5] M.J. Baines, Moving Finite Elements, Oxford University Press, (1994).

[6] Y. Tourigny and M.J. Baines, Analysis of an algorithm for generating locally

optimal meshes for L2

approximation by discontinuous piecewise polynomials,

 Math. Comp., 66, 623–650, (1998).

[7] P.L. Roe, Compounded of many simples. In Proceedings of Workshop on Bar-

riers and Challenges in CFD, ICASE, NASA Langley, August 1996, Ventakr-

ishnan, Salas and Chakravarthy, eds., 241-, Kluwer (1998).

[8] M.J. Baines and S.J. Leary, Fluctuation and signals for scalar hyperbolic equa-

tions on adjustable meshes, Com. Num. Meth. Eng., 15, 877–886, (1999).

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[9] M.J. Baines, Algorithms for optimal discontinuous piecewise linear and con-

stant L2 fits to continuous functions with adjustable nodes in one and two

dimensions, Math. Comp., 62, 645–669, (1994).

[10] M. Delfour et al., An optimal triangulation for second order elliptic problems,

Comput. Meths. Applied Mech. Engrg., 50, 231–261, (1985).

[11] K. Miller and M.J. Baines, Least Squares MovingFinite Elements, OUCLreport

98/06, Oxford University Computing Laboratory, (1998), to appear in the IMA

 Journal of Numerical Analysis.

[12] G.F. Carey and H.T. Dinh, Grading functions and mesh distribution, SIAM J.

 Num. An., 22, 1028–1050, (1985).

[13] H. Deconinck, P.L. Roe, and R. Struijs, A multidimensional generalisation of 

Roe’s flux difference splitter for the Euler equations, Computers and Fluids,

22, 215, (1993).

[14] S.J. Leary, Least Squares Methods with Adjustable Nodes for Steady Hyperbolic

PDEs, PhD Thesis, Department of Mathematics, University of Reading, UK,

(1999).

[15] M.J. Baines, S.J. Leary, and M.E. Hubbard, Multidimensional least squaresfluctuation distribution schemes with adaptive mesh movement for steady hy-

perbolic equations, (2000) (submitted to SIAM J. Sci. Stat. Comp.,). See also

by the same authors A finite volume method for steady hyperbolic equations,

in Proceedings of Conference on Finite Volumes for Complex Applications II,

R. Vilsmeier, F. Benkhaldoun and D. Hanel, eds., Duisburg, July 1999, 787–

794, Hermes, (1999).

[16] J.Y. Trepanier, M. Paraschivoiu, M. Reggio, and R. Camarero, A conservative

shock fitting method on unstructured grids, J. Comp. Phys., 126, 421–433,(1996).

[17] L.M. Mesaros and P.L. Roe, Multidimensional fluctuation splitting schemes

based on decomposition methods, Proceedings of the 12th AIAA CFD Confer-

ence, San Diego, (1995).

[18] H. Yee, R.F. Warming, and A. Harten, Implicit total variation diminishing

(TVD) schemes for steady state calculations, J. Comp. Phys., 57, 327–366,

(1985).

[19] M.J. Baines and M.E. Hubbard, Multidimensional upwinding with grid adap-

tation, in Numerical Methods for Wave Propagation, E.F. Toro and J.F. Clarke,

eds., Kluwer, (1998).

[20] M.J. Baines, Least-squares and approximate equidistribution in multidimen-

sions, Numerical Methods for Partial Differential Equations, 15, 605–615,

(1999).

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[21] H. Nishikawa, The Discrete Least Squares Method for 2D Stokes Flow, Tech-

nical Report (unpublished), Department of Aerospace Engineering, University

of Michigan, (1997).

[22] E.A. Dorfi, and L.O’C. Drury, Simple adaptive grids for 1D initial value prob-

lems, J. Comput. Phys., 69, 175–195, (1987).

[23] W. Huang and R.D. Russell, Moving mesh strategy based upon a gradient flow

equation for two-dimensional problems, SIAM J. Sci Stat. Comput., 20, 998,

(1999).

[24] D. Ait-Ali-Yahia, W.G. Habashi, A. Tam, M.G. Vallet, and M. Fortin, A direc-

tionally adaptive finite element method for high speed flows, Int. J. for Num.

 Meths. in Fluids, 23, 673–690, (1996).

[25] J.A. Mackenzie, A Moving Mesh Finite Element Method for the Solution of 

Two-Dimensional Stefan Problems, Technical Report 99/26, Department of 

Mathematics, University of Strathclyde, UK, (1999).

[26] C. Johnson, Finite Element Methods for Partial Differential Equations, Cam-

bridge University Press, (1993).

[27] M.E. Hubbard, Multidimensional Upwinding and Grid Adaptation for Conser-

vation Laws, PhD Thesis, Department of Mathematics, University of Reading,UK, (1996).

[28] H. Nishikawa, M. Rad, and P.L. Roe, Grids and Solutions for Residual Distri-

bution, Private communication, (2000).

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Chapter 9

 Linearly Implicit Adaptive Schemes forSingular Reaction-Diffusion Equations

Q. Sheng and A.Q.M. Khaliq

9.1 Introduction

Many important physical processes, such as the combustion of gases in a heat

engine, can be described by nonlinear reaction-diffusion partial differential equations

with singular or near-singular source terms. The rate of change of the solution of 

such equations may blow up in finite time, while the solution itself remains bounded,

when certain physical quantities, such as the size of the combustor, reach their limits.

The phenomenon is often referred to as quenching [1], [4]–[7], [11]–[13], [26]–[30],

[32].

Mathematically, quenching phenomena can be interpreted as the blow-up of rates

of change of solutions of nonlinear reaction-diffusion differential equations. This can

occur when certain physical parameters, for example, the size of the combustor, reachtheir critical limits when particular gases are used. It has become extremely important

to estimate such limit values efficiently and effectively for various reaction-diffusion

models so that better controls and designs can be achieved in industrial applications.

Consider the following simplified reaction-diffusion problem with a highly non-

linear source function:

ut  = uxx + f (u), 0 < x < a, 0 < t < T , (9.1)

u(x, 0)=

u0

, 0 < x < a;

u(0, t)=

u(a,t)=

0, 0 < t < T, 0≤

u0

< 1 , (9.2)

where f(u) = 1/(1 − u)θ , θ > 0, T  ≤ ∞, and a is an important parameter playing

in the combustion process. This model describes a steady-state combustion of two

gases meeting in a gap between porous walls at distance a apart. Fuel diffuses at one

wall, oxidant at the other with a zone of reaction between the walls, and dying out

towards each wall. Here x is the distance from one of the walls, t  is the time, and u is

the uniformly scaled temperature. Further, θ  is a physical property index of the gases

involved [2, 12, 13]. Kawarada [17] discovered in 1975 that when θ  = 1 there exists

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for solving quenching-type singular reaction-diffusion equations are difficult and are

not fulfilled until recent studies of Sheng, Khaliq, and Cheng [26]–[29].

We will consider a more general degenerative reaction-diffusion model,

xq ut  = uxx + c(x)ux + f (u), 0 < x < a, 0 < t < T , q ≥ 0 , (9.3)

together with initial and boundary conditions (9.2). Discussions of the existence and

uniquenessof its solutions canbe found in [4, 7, 21] and references therein. Numerical

techniques from the method of fundamental solutions, finite element approximations,

and Douglas algorithms used for solving the equation are investigated by several

authors (see [5, 6] and references therein). Most of the approaches, however, are

still indirect and considerations of reduced problems of (9.3) are required. Less

complicated, more efficient adaptive numerical methods have become necessary forcomputations of quenching solutions and critical values. This becomes the goal of 

our discussion.

Let y = x/a. Equation (9.3), together with (9.1), can be reformulated as

yq ut  =1

aq+2uyy + c(ay)

aq+1uy + f (u), 0 < y < 1, 0 < t < T , (9.4)

u(ay, 0) = u0, 0 < y < 1;

u(0, t) = u(a,t) = 0, 0 < t < T, 0 ≤ u0 < 1 , (9.5)

where f  = a−q f . A computational advantage of the above reformulation is that the

discretization in space becomes simpler. We avoid dealing with a very sensitive a

in the discretization, and move the quenching parameter directly into the differential

equation. This will be helpful for introducing proper adaptive mechanisms later. We

also note that in the particular case when c ≡ 0 and a ≥ 1, (9.4) reduces to the

standard parabolic equation with the singular source term, that is,

yq

ut  = uyy + f (u), 0 < y < 1, 0 < t < T , (9.6)

where 0 < = 1/aq+2 ≤ 1.

In this study, we will construct efficient adaptivemethods for computing the numer-

ical solution, critical length, and quenching time of the nonlinear reaction-diffusion

problem (9.4) and (9.5) directly. Nonlinear source functions with different indices θ 

will be considered in the numerical demonstrations. Techniques of semidiscretiza-

tions in spatial variables are used. For the system of nonlinear ordinary differen-

tial equations obtained, we introduce a two-stage Runge–Kutta solver, then L-stable

rational functions with real and distinct poles for approximating derived matrix ex-

ponentials. The modified arc-length adaptive mechanism is established. Special

consideration is given to the stability and efficiency in handling the degenerate and

singular properties. The semi-adaptive method constructed is of second-order accu-

racy, while the fully adaptive scheme is of first-order accuracy. We then compare

our results with existing results obtained by Acker and Walter [1], Chan et al. [5],

and Walter [32], and show that our numerical method is accurate and reliable. An

important feature of our algorithm may be that it does not depend on the structure

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of the nonlinear source term presented. The numerical scheme can thus be extended

for solving more sophisticated reaction-diffusion models with different inputs and

sources, and for solving multidimensional problems without major difficulties.

9.2 The Semi-Adaptive Algorithm

9.2.1 The Discretization

Let q

≡0. For positive integer N , we define h

=1/(N 

+1) as the spatial dis-

cretization parameter and let uk(t) be an approximation of the exact solution of (9.4)

and (9.5) at (hk,t), k = 0, 1, . . . , N  + 1. Replacing the first- and second-order

spatial derivatives in (9.4) by central difference approximations

∂v(y,t)

∂y= v(y + h,t) − v(y − h,t)

2h+ O(h2) ,

∂2v(y,t)

∂y 2= v(y + h,t) − 2v(y,t) + v(y − h,t)

h2+ O(h2) ,

respectively, we may formulate the approximation of (9.4) and (9.5) as an initial value

problem for the unknown function v through the method of lines. Namely,

vt (t) = Av(t) + g(v(t)), 0 < t < T , (9.7)

v(0) = v0 , (9.8)

where v(t) = (u1(t),u2( t ) , . . . , uN (t))T , g(v(t)) = (f (u1(t)), f (u2(t)),... ,

f (uN (t)))T  and v0

=(u0(t 1), u0(t 2) , . . . , u0(t N ))T . The matrix A is generated from

the semidiscretization process. It is nonsingular in most cases, and is symmetric andnegative definite when c ≡ 0 according to the central difference approximation used.

The formal solution of (9.7) and (9.8) can be expressed as

v(t) = E(tA)v0 + t 

0

E((t − τ)A)g(v(τ))dτ, 0 < t < T , (9.9)

where E(ξQ) = exp(ξQ) is the analytic semigroup generated.

Formula (9.9) indicates good chances to construct highly efficient time integrators

for solving systems of ordinary differential equations, regardless of any particular

spatial discretizationadopted. For instance, we may consider certain adaptiveRunge–

Kutta methods, in which variable time steps are generated through proper arc-length

mechanisms. The matrix exponential operators obtained can be approximated by

L-stable rational approximations [18, 31]. A consistent compound adaptive method

may be developed based on the above considerations to assure an accuracy in the

computation that will not be affected by the existing singularities. The computation of 

the numerical solution may well reflect the special feature of quenching singularities.

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Let inequalities between vectors be in the componentwise sense. We consider the

numerical analog of the formal solution (9.9) through the adaptive two-stage Runge–

Kutta integrator:

w(1) = v0 , (9.10)

w(2) = R(2)0 (τA)v0 + τ R

(2)1 (τA)g1 , (9.11)

w(3) = R(3)0 (τA)v0 + τ 

κR

(3)1 (τA) + (1 − 2κ)R

(3)2 (τA)

g1

+

(1 − κ)R(3)1 (τA) + (2κ − 1)R

(3)2 (τA)

g2

, (9.12)

v1 = w(3) , (9.13)

where 0 ≤ κ ≤ 1,

gk = g

w(k)(τ )

, k = 1, 2 , (9.14)

R(i)j  (z) =

R

(i)j −1(z) − I 

z−1, i = 2, 3, j  = 1, 2 , (9.15)

and R(i)0 , i = 2, 3, are proper approximations to E. The temporal discretization

parameter, τ , will be determined through a properly defined adaptive mechanismduring the computation. Note that functions R

(i)j  (z), j  = 1, 2, possess the same

denominator as R(i)0 (z). In fact, the factor z−1 can be canceled out during actual

computations if R(i)0 , i = 2, 3, are properly chosen. This implies that the constructed

algorithm is valid even when A is singular. The linearized formula offers a direct way

for computing solutions of (9.1) through (9.3). The Runge–Kutta time integrator is

stable when the real parts of the eigenvalues of A are negative.

Our purpose is to derive a method of consistency order two. To this end, at the same

time of adopting the above Runge–Kutta process, we use an L-stable second-orderrational approximation R

(i)0 , i = 2, 3, for E.

9.2.2 The Adaptive Algorithms

For the algorithm (9.10) through (9.15), we consider rational approximations to

E, in particular the second-order L-stable approximation with distinct real poles

(see [18, 31] for details),

R(z) = w1

1 − b1z+ w2

1 − b2z,

where b1 = 1/9, b2 = 1/3, w1 = −8, w2 = 9. By letting A1 = I  − τ 4 A, A2 =

I  − τ 3

A, τ > 0, based on the function R, we may define

R(i)0 (τA) = −8A−1

1 + 9A−12 =

I  + 5τ 

12A

A−1

1 A−12 , i = 2, 3 . (9.16)

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It follows from (9.15) and (9.16) that

R(2)1 (τA)

= −2A−1

1

+3A−1

2

=

1

12

(12I 

−τA)A−1

1 A−12 , (9.17)

R(3)1 (τA) = R

(2)1 (τA) , (9.18)

R(3)2 (τA) = −1

2A−1

1 + A−12 = 1

12(6I  − τA)A−1

1 A−12 . (9.19)

Let t k = k τ , k = 0, 1, . . . , and vk = v(t k), k = 1, 2, . . . , be numerical solutions

of (9.7) and (9.8) obtained by using the two-stage method (9.10) through (9.15). We

denote

w

(1)

= vk, k = 0, 1, . . . , (9.20)w(2) = R

(2)0 (τA)w(1) + τ R

(2)1 (τA)g1 . (9.21)

Then the numerical solution can be formulated in the following embedded form,

vk+1 = R(3)0 (τA)w(1) + τβ , (9.22)

where v0 is the initial value and

β

= κR(3)1 (τA)

+(1

−2κ)R

(3)2 (τA) g1

+ (1

−κ)R

(3)1 (τA)

+ (2κ − 1)R(3)2 (τA)

g2 .

Substituting (9.17) through (9.19) into (9.20) through (9.22), we obtain readily that

w(1) = vk, k = 0, 1, . . . , (9.23)

A1A2w(2) =

I  + 5τ 

12A

w(1) + τ 

12(12I  − τA)g1 , (9.24)

A1A2vk+1 = I  +5τ 

12 Aw(1)

+ τ γ , (9.25)

where γ  = A1A2β. Since A is a tridiagonal matrix, thus A1A2 is of quindiagonal.

However, this will not add any extra cost in solving (9.23) through (9.25). In fact, we

may observe that by denoting

p =

I  + 5τ 

12A

w(1) + τ 

12(12I  − τA)g1 ;

q = I  +5τ 

12 Aw(1)

+ τ γ ,

systems of linear equations (9.23) through (9.25) can be reformulated as

w(1) = vk, k = 0, 1, . . . , (9.26)

A1y(1) = p, A2w(2) = y(1) , (9.27)

A1y(2) = q , (9.28)

A2vk+1 = y(2), k = 0, 1, . . . , (9.29)

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which is of clearly tridiagonal. We have the following:

 LEMMA 9.1 Let c(x), 0 ≤ x ≤ 1 , be continuous and let σ  = max0≤x≤a |c(x)|. If (i) σ  = 0 , or 

(ii) h ≤ 2/σ  and (6h2 + 2τ)/hτ  ≥ c(ah), −c(aNh), σ  = 0 ,

then matrices A1, A2 are monotone and thus nonsingular. Their inverses are positive

and monotone.

PROOF Let a(1)i,j  , a(2)

i,j  , i , j  = 1, 2, . . . , N  , be elements of A1, A2, respectively. It

is not difficult to see that both A1, A2 are irreducible. We only need to give a detailed

proof regarding A1 since the proof of the case for A2 will be similar. The case when

σ  = 0 is obvious. Therefore, we may assume that σ  = 0 throughout our discussion.

We observe that for nontrivial elements of A1 we have

a(1)i,i = 1 + τ/2h2, i = 1, 2, . . . , N  ; (9.30)

a(1)i,i

+1

= −τ (2

+hc(aih))/8h2, i

=1, 2, . . . , N  

−1

;(9.31)

a(1)i+1,i = −τ (2 − hc(aih))/8h2, i = 2, 3, . . . , N . (9.32)

To meet the criteria a(1)i,j  ≤ 0, i = j , it is necessary to have that 2 ± hc(aih) ≥ 0,

i = 1, 2, . . . , N  − 1, [14]. These imply that

1

h≥ c(aih)

2,

1

h≥ −c(aih)

2, i = 1, 2, . . . , N . (9.33)

Further, it is found thatN 

j =1

a(1)i,j  = 1, i = 2, 3, . . . , N  − 1 ,

and

N 

j =1

a(1)1,j  = 1 + τ 

4h2− τc(ah)

8h;

N j =1

a(1)N,j  = 1 + τ 

4h2+ τc(aNh)

8h.

To let the above sums be nonnegative, respectively, it is necessary that

1 + τ 

4h2− τc(ah)

8h, 1 + τ 

4h2+ τc(aNh)

8h≥ 0 . (9.34)

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Together with (9.33), condition (9.34)ensures that A1 ismonotoneand A−11 is positive

and thus monotone.

As for A2, similarly, for nontrivial elements we have

a(2)i,i = 1 + 2τ/3h2, i = 1, 2, . . . , N  ;

a(2)i,i+1 = −τ (2 + hc(aih))/6h2, i = 1, 2, . . . , N  − 1 ;

a(2)i+1,i = −τ (2 − hc(aih))/6h2, i = 2, 3, . . . , N .

It follows that A2 and A−12 are monotone if (9.33) and

1 +τ 

3h2 −τc(ah)

6h , 1 +τ 

3h2 +τc(aNh)

6h ≥ 0 . (9.35)

Inequality (9.33) suggests the first condition in (ii), and a combination of (9.34) and

(9.35) gives the rest of constraints. Hence the proof is completed.

Further, we may prove the following.

THEOREM 9.1

Given 0 ≤ u0 << 1 and 0 ≤ κ ≤ 1 , let c(x), 0 ≤ x ≤ 1 , be continuous and let σ  = max0≤x≤a |c(x)|. If (i) σ  = 0 , or 

(ii) h ≤ 2/σ,(6h2 + 2τ)/hτ  ≥ c(ah), −c(aNh) and τ/ h2 ≤ 6/5, σ  = 0 ,

then solution vectors of  (9.26) through (9.29) , {vk}∞k=0 ,

(1) form a monotonically increasing sequence;

(2) increase monotonically till unity is exceeded by an element of the solution vector,

or converges to the steady solution of the problem (9.7) and (9.8).

 In the latter case, we do not have a quenching solution.

PROOF The theorem is obvious when σ  = 0. Assuming σ  = 0, we first consider

the vector p. It is not difficult to see that nontrivial entries of the matrix function

B

=I 

+5τ 12 A are

bi,i = 1 − 5τ/6h2, i = 1, 2, . . . , N  ;bi,i+1 = 5τ (2 + hc(aih))/24h2, i = 1, 2, . . . , N  − 1 ;bi+1,i = 5τ (2 − hc(aih))/24h2, i = 2, 3, . . . , N .

Therefore, B > 0 if 

τ/ h2 ≤ 6/5, h ≤ 2/|c(aih)|, i = 1, 2, . . . , N . (9.36)

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At the same time, we observe that

v0 ≤ Bv0 + τ 

12(12I  − τA)g1 = p ≤ A−1

1 p = y(1) ≤ A−12 y(1) = w(2)

under condition (ii). Recall (9.26). From the above and (9.38), inequalities (9.37)

become obvious.

It can be shown that γ  ≥ 0. To see this, we set γ  = 1g1 + 2g2, where

1 = 1

12(6I  + τ (κ − 1)A), 2 = 1

12(6I  − τκA), 0 ≤ κ ≤ 1, τ > 0 .

Recall the equality A = 4τ 

(I  − A1) and relations (9.30) through (9.32). We find

immediately that 1, 2 > 0 and it follows that γ 

≥0.

Based upon the previous discussions, we may conclude that

0 ≤ v0 = w(1) ≤ Bw(1) <

I  + 5τ 

12A

w(1) + τ γ  ≤ A−1

1 qi

= y(2) ≤ A−12 y(2) = v1 ,

for the positivity of  A−11 , A−1

2 , and B. Next, by replacing w(1), w(2) with more

general notations w(1,k) , w(2,k), respectively, and g1, g2 by g1,k, g2,k , respectively, in

the computation of uk+1 from uk , subsequently we obtain that

A1A2 (vk+1 − vk) =B (vk − vk−1)

+ τ A−11 A−1

2

1(g1,k − g1,k−1) + 2(g2,k − g2,k−1)

,

k =1, 2, . . . .

Recall that v1 −v0 > 0, gj,1 −gj,0 > 0, j  = 1, 2, and the fact that A1A2 is monotone

and j , j  = 1, 2, are positive under conditions given by the theorem. An inductive

argument leads immediately to

v0 < v1 < v2 < v3 < · · · < vk < · · · < 1 ,

if kτ < T a and the sequence exceeds unity if kτ  ≥ T a by Nagumo’s lemma [1].

It may be interesting to see the constraints in h and τ . The restrictions are necessary

to guarantee the monotonicity required for approximating the solution of nonlinear

quenching problems, as discussed in various publications (see [27, 28], for instance).

Now we consider a modified arc-length adaptive mechanism in time. Let ut (hk,t)

be the time derivative of the solution of (9.4) and (9.5) at (hk,t),k = 0, 1, . . . , N  +1.When t − t > 0, 0 < t << 1, the arc-length of the function ut  between (hk, t −t),(hk,t) can be approximated by

(t)2 + (ut (hk,t) − ut (hk,t − t))2

12 . Let

τ < 1 be the given initial time step size, and τ k be the time step reference for

determining the actual time step size to be used in the next step computation. We

require that  (t)2 + (ut (hk,t) − ut (hk,t − t))2

t = τ 

¯τ k

.

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It follows immediately that

¯τ k

=τ t 

 (t)2 + (ut (hk,t) − ut (hk,t − t))2

= τ  1 +

ut (hk,t)−ut (hk,t −t)

t 

2≤ τ . (9.39)

It is observed that when t  → 0, we have

τ k → τ 

 1 + u2t t (hk, t)

which is similar to the monitoring function used in traditional adaptive algorithms [9,

10]. However, ourarc-lengthadaptivemechanismis established based on the function

ut  rather than u.

In practical computations, function values of  ut  can be conveniently obtained

through Equation (9.4), together with proper difference approximations. Instead

of sophisticated smoothing processes, we may introduce a minimal time step size

controller τ 0, 0 < τ 0 << τ , to avoid unnecessarily large numbers of computations

immediately before the blow-up of ut  and to indicate a proper stopping time for the

computation. Under the consideration, the actual time step size used for computing

the solution at a higher time level, u(hk, t + t), can be determined uniquely by the

following formula,

t  = max

τ 0, min

k{τ k}

. (9.40)

9.3 The Fully Adaptive Algorithm

9.3.1 The Discretization

We now rewrite (9.4) and (9.5) into the following uniformed form:

ut  =1

aq+2yq uyy +c(ay)

aq+1yq uy +1

aq yq f (u),

0 < y < 1, 0 < t < T , (9.41)

u(ay, 0) =u0, 0 ≤ y ≤ 1; u(0, t) = u(a,t) = 0,

0 < t < T, q ≥ 0 . (9.42)

Given N > 0. We define a nonuniform partition N  over the interval [0, 1] : N  ={y0, y1, . . . , yN +1}, where y0 = 0, yj  = yj −1 + hj , hj  > 0, j  = 1, 2, . . . , N  +

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1, and yN +1 = 1. Further, based on N , we introduce the following difference

approximations for the first and second order derivatives in space, respectively,

D+wj  =wj 

+1

−wj 

hj +1, j  = 1, 2, . . . , N  ; (9.43)

D+D−wj  =1

hj +1hj 

wj +1 − 2

hj hj +1wj  +

1

hj hj 

wj −1,

j  = 1, 2, . . . , N , (9.44)

where hj  = (hj  + hj +1)/2 and wj  = w(yj , t). Denote uj (t) as an approximation of 

the exact solution of (9.1) and (9.2) at the grid point (yj , t ) , j  = 0, 1, . . . , N  +1, and

let v(t)=

(u1(t),u2( t ) , . . . , uN (t))T . It follows that, by replacing the spatial deriva-

tives with the above differences and removing higher order truncation error terms, we

arrive at the following system of nonlinear semidiscretized equations corresponding

to (9.1) and (9.2):

vt (t) = Av(t) + g(v(t)), 0 < t < T , (9.45)

v(0) = v0 , (9.46)

where g(v(t)) = (f (u1(t)), f (u2( t ) ) , . . . , f ( uN (t)))T , v0 = (u0(y1), u0(y2) , . . . ,

u0(yN ))T , and A

∈R

N ×N  is nonsingular. The order of accuracy of (9.2) and

(9.3) is of one unless hj  = h > 0, j  = 1, 2, . . . , N  + 1, in which we have theorder two. Similar to (9.10) through (9.13), a discrete analog of the abstract solution

formula (9.9) can be given via the two-stage second-order adaptive Runge–Kutta

method by the following:

w(1)k := vk , (9.47)

w(2)k := R

(2)0 (τ kA)w

(1)k + τ kR

(2)1 (τ kA)g1 , (9.48)

w(3)k

:=R

(3)0 (τ kA)w

(1)k

+τ k κR

(3)1 (τ kA)

+(1

−2κ)R

(3)2 (τ kA) g1

+

(1 − κ)R(3)1 (τ kA) + (2κ − 1)R

(3)2 (τ kA)

g2

, (9.49)

vk+1 := w(3)k , k = 0, 1, . . . , K , (9.50)

where vk is the approximation of vk, 0 ≤ κ ≤ 1, τ k > 0. Functions gi , R()j  (z),i,j  =

1, 2, = 2, 3, are defined through (9.14) and (9.15). The algorithm offers a possible

access for computing the solution of (9.41) and (9.42) through moving grid in time.

The stability of the Runge–Kutta time integrator is again guaranteed when real parts

of the eigenvalues of A are nonpositive. The temporal discretization parameter, τ k ,will be determined through our modified arc-length adaptive mechanism during the

computation.

Let ∧ be one of the operations <, ≤, >, ≥, and α, β ∈ RN . We introduce the

following notations:

1. α ∧ β means αi ∧ βi , i = 1, 2, . . . , N  ;

2. α ∧ a means αi ∧ a, i = 1, 2, . . . , N  , for any given a ∈ R.

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9.3.2 The Monotone Convergence

Consider the following second-order L-acceptable rational approximation with

distinct real poles (see [29, 31] for details),

R(z) = 1 + a1z

(1 − b1z)(1 − b2z),

where

b1 + b2 + a1 = 1 ,

b1 + b2 − b1b2 = 1

2,

a1, b1, b2 > 0 .

Similar to (9.16) through (9.19), by denoting t k = Kk=0 τ k , and letting vk = v(t k)

be the numerical solution of (9.45) and (9.46) obtained through the adaptive pro-

cedure (9.47) through (9.50) at stage t k , we may show that the numerical solution

satisfies the following procedure:

w(1)k = vk , (9.51)

w

(2)

k = R

(2)

0 (τ kA)w

(1)

k + τ kR

(2)

1 (τ kA)g1 . (9.52)vk+1 = R

(3)0 (τ kA)w

(1)k + τ kβk , (9.53)

where v0 is the initial vector from (9.46) and

βk =

κR(3)1 (τ k A) + (1 − 2κ)R

(3)2 (τ kA)

g1

+

(1 − κ)R(3)1 (τ kA) + (2κ − 1)R

(3)2 (τ kA)

g2 .

We subsequently obtain that

w(1)k = vk , (9.54)

A1A2w(2)k = (I  + a1τ kA)w

(1)k + τ k(I  − b1b2τ kA)g1 , (9.55)

A1A2vk+1 = (I  + a1τ kA)w(1)k + τ kA1A2βk, k = 0, 1, . . . , K . (9.56)

Note that A is of tridiagonal. In fact, for nonzero elements of A = [ai,j ]i,j =1,...,N ,

we may show that

aj,j  = − 2 + ahj c(ayj )

aq+2y

qj hj hj +1

−1, j  = 1, 2, . . . , N  ;

aj,j +1 =

1 + ahj c(ayj )

aq+2yqj hj +1hj 

−1, j  = 1, 2, . . . , N  − 1 ;

aj,j −1 =

aq+2yqj  hj hj 

−1, j  = 2, 3, . . . , N .

It follows immediately that A1A2 is of quindiagonal.

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Define

pk = (I  + a1τ kA) w(1)k + τ k(I  − b1b2τ kA)g1 ,

qk = (I  + a1τ kA) w(1)k + τ kA1A2βk .

Systems (9.54) through (9.56) can be then simplified into the following:

w(1)k = vk , (9.57)

A1ξ (1)k = pk, A2w

(2)k = ξ (1) , (9.58)

A1ξ (2)k = qk, A2vk+1 = ξ 

(2)k , k = 0, 1, . . . , K . (9.59)

 LEMMA 9.2

 Let function c(x) be continuous on [0, a] , and σ  = max0≤x≤a |c(x)|. If (i) σ  = 0 , or 

(ii) hj  < 1σ a

, j  = 1, 2, . . . , N  + 1, σ  = 0 ,

then the real parts of the eigenvalues of A are nonpositive. Further, at least one of 

the real parts is negative.

PROOF The proof is straightforward according to Gerschgorin theorem [16].

 LEMMA 9.3

 Let function c(x) be continuous on [0, a] , and σ  be the same as defined in Lemma 9.2.

 If 

(i) σ  = 0 , or 

(ii) hj  < 1σ a

, j  = 1, 2, . . . , N  + 1, σ  = 0 ,

then the matrices A1, A2 are monotone and nonsingular. Their inverses are positive.

PROOF The properties can be shown directly according to the structures of the

matrices A1, A2.

 LEMMA 9.4

 Let function c(x) be continuous on [0, a] , and σ  = max0≤x≤a |c(x)|. If (i) σ  = 0 , or hj  < 1

σ a, j  = 1, 2, . . . , N  + 1, σ  = 0 ,

(ii) 0 ≤ w(1), w(2) < 1 ,

(iii) Aw(1) + g(w(1)) > 0 ,

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then vk = w(1) < w(2) < vk+1. Thus the sequence {vk} is monotonically increasing.

PROOF Recalling the definition of A1, A2, we have

A1A2

w(2) − w(1)

= τ k(I  − b1b2τ kA)

Aw(1) + g

w(1)

,

A1A2

vk+1 − w(2)

= τ k

1

2I  − τ kκb1b2A

(g2 − g1) .

It follows that

A−11 (I 

−b1b2τ k A)

=(1

−b2)A−1

1

+b2I > 0, 0 < b2 <

1

2

,

A−11

1

2I  − τ k κb1b2A

=

1

2− κb2

A−1

1 + κb2I > 0 .

Thus, by previous lemmas, we obtain immediately that vk = w(1) < w(2) < vk+1

and this shows the required monotonicity.

 LEMMA 9.5

 Let function c(x) be continuous on [0, a] , and let σ  = max0≤x≤a |c(x)|. If (i) σ  = 0 , or hi < min

1

σ a, 1

Maq+2

, i = 1, 2, . . . , N  + 1 ,

(ii) h1h2, hN −1hN  < 12ξ aq+2 ,

(iii) there exists a constant c1 with 0 < c1 < 1 such that 

1

a2q

+4

y

2q

1 h1h2h1h2

,1

a2q

+4

y

2q

N −1hN −1hN hN hN +1

<c1

τ 

2

0

,

τ 0 ≤ min

1

2ξ (1 − c1),

2

M + ξ (1 − c1)

,

where ξ  = f (0) and M  = f (1/2) , then for any null vector w(1) we have w(2) <

1/2, v1 < 1.

PROOF Let w

=(1, 1, . . . , 1)T . We first show that w(2) < 1/2 under the

conditions given. Denote

s = A1A2

1

2w − w(2)

= 1

2A1A2w − (I  + a1τ 0A)w(1) − τ 0(I  − b1b2τ 0A)g1

=

1

2− ξ τ 0

I  − b1 + b2

2τ 0A + 1

2b1b2τ 20 A(2ξ I  + A)

w .

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By means of the above lemmas, we have

THEOREM 9.2 Let function c(x) be continuous on [0, a] , and let σ  = max0≤x≤a |c(x)|. Assume that 

{vk}∞k=0 be a solution sequence given by the two-stage scheme (9.57) through (9.59).

 If 

(i) σ  = 0 , or hj  < 1σ a

, j  = 1, 2, . . . , N  + 1, σ  = 0 ,

(ii) Av + g(v) > 0 for v < 1 ,

then

{vk

}∞k

=0

(1) forms a monotonically increasing sequence,

(2) increases monotonically until unity is exceeded by an element of the solution

vector, or converges to the steady solution of the problem (9.7) and (9.8).

 In the later case, we do not have a quenching solution.

The adaptive grid distribution over the interval [0, 1] is determined by a modified

arc-length adaptive principal. As stated before, the interval can be partitioned through

{yj , j  = 0, 1, . . . , N , N  + 1 : y0 = 0, . . . , N  }, can be obtained via following grid

equations  yj +1

yj 

M(x,t)dx = 1

N 

 1

0

M(x,t)dx, 0 ≤ j  ≤ N ,

under a proper smoothness process. The adaptation in time can be achieved through

an approach similar to [26].

9.3.3 The Error Control and Stopping Criterion

It has been essential to estimate the computational error development during the

calculation. The information obtained is not only used for determining the time

to stop, but also for optimizing the computation procedures. Practically, an error

estimate formula for monitoring the local relative error at each time step is widely

adopted. In this chapter, weconsider a Milne-alikedevice forachieving this. Theerror

assessment obtained is then used to help updating temporal discretization parameters

τ k and constructing a reasonable stopping criteria.

There are two frequently used error controlling strategies over the local error for a

given tolerance > 0. One is the error control per step, while the other is the error 

control per unit step including the standard Milne device [16]. Most of the traditional

methods, including the Milne device, are not suitable to be used directly for adaptive

time-step methods due to the fact that they require executing two numerical methods

at the same time over a nonuniform grid.

Based on the fact that nonuniform grids spread both in the time and space di-

rections in our applications, we adopt a Milne-alike mechanism for the local error

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estimate. Let uk be thenumerical solution of (9.41)and (9.42)obtained through (9.57)

through (9.59) at the time level t k, 0 < k ≤ K. Then the local error for computing

uk

+1 at t k

+1

=t k

+τ  is defined as

u(t k + τ ) − uk+1 = τ p+1(t k, uk) + O

τ p+2 , 0 < k ≤ K , (9.60)

where p > 0 is the order of accuracy, is the principal error function, and u(t) is

the exact solution of (9.41) and (9.42).

Now, instead of τ , we choose τ/2 and repeat the computation. It follows that for

the new solution uk+1 at t k + τ  we have

u(t k+

τ )− ˜

uk+

1

=c

τ 

2p+1

(t j , uj )+

O(τ p+2) , (9.61)

where for the positive constant c, c = O(1). Subtracting (9.61) from (9.60), we

readily find that

τ p+1(t k, uk) = uk+1 − uk+1

1 − c2p+1

+ O(τ p+2) .

Therefore, it follows

u(t k + τ ) − uk+1 = 11 − c

2p+1

(uk+1 − uk+1) + O τ p+2≈ 1

1 − c2p+1

(uk+1 − uk+1) , 0 < k ≤ K . (9.62)

Given δ > 0. An error control per unit step approach is to require the local error κ

satisfies

κ ≤ τ δ ,

where

κ = 2p+1

2p+1 − c

uk+1 − uk+1

originates in (9.62). By evaluating κ, we decide if uk+1 is an acceptable approxima-

tion. If not, the computed solution at time step k + 1 is rejected: we go back to uk

and pick a smaller τ . Moreover, if κ is significantly smaller than τ δ, we take this

as an indication that the time step is too small and may be increased. Based on the

above criterion, we may further predict a suitable step size to be used in the next stepcomputation. Note that the local error of a next step solution will follow:

u (t k+1 + τ new) − uk+2 = τ p+1new (t k+1, uk+1) + O

τ 

p+2new

.

According to (9.62), we may assume that uk+1 = uk + O(τ). An appropriate

differentiability of the principal error function suggests the estimate

(t k+1, uk+1) = (t k, uk ) + O(τ ) .

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With the observation, we predict that

τ new =

τ δ

κ1/(p+1)

.

An actual time step to use is determined through a combination of the above error

control per unit step criterion and a modified arc-length mechanism. The combination

can also help in building a proper stopping criterion for the quenching computation.

It is observed that while quench has not occurred, the numerical solution via (9.57)

through (9.59) converges and the computational error is mainly contributed from the

truncation error and remains smooth. When the quench is about to occur, however, the

computational error changes dramatically, especially when t k is sufficiently close to

the quenching time. Any standard control criterion may break down during this stage.

The reasons are as follows. (1) As t k is getting closer and closer to the quenching

time, a sharp change in the derivative of the physical solution u starts. This demands

tinier and tinier step sizes to be acquired and used according to a standard error

feedback controller. The demand soon becomes impractical due to the increasing of 

the computational cost and rounding error; thus, the controller fails. (2) The physical

solution breaks down at the quenching time and becomes undefined. A numerical

solution becomes unsteady, or blows-up, near the quenching point and does not make

any sense at that point. This may generate an uncontrollable error.The actual stopping criterion we considered is as follows:

1. If uk∞ ≥ 1, then we denote t k−1 as the computational quenching time and stop

the computation;

2. Let rk = ek/ek−1 be the error ratio, where ek = uk − uk∞ is the error reference

at time t k . If rk > λ where λ is a controlling constant determined through the

combination of the aforementioned error analysis and the arc-length criterion,

λ 1, then denote t k as the computational quenching time and stop thecomputation.

3. Otherwise, stop when t k = T .

9.4 Computational Examples and ConclusionsIt has never been an easy task to approximate numerically critical values of a

quenching problem. Our second-order accurate adaptive algorithm (9.26) through

(9.29) provides a reliable way for solving the nonlinear partial differential equa-

tions (9.3). The compound structure of the adaptive scheme is relatively simple and

takes advantage of several known computing techniques. Without loss of generality,

the initial value u0 is set to be zero. κ = 0 is considered. The spatial mesh step size

is chosen as 0.1, while the initial time step varies from 0.01 to 0.001. The purpose

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of choosing a smaller initial time step size is not for the stability of the numerical

scheme, but for observing more accurately the quenching behavior. When numerical

solutions are advanced to be near the quenching point, should quenching exist, they

become very sensitive and the rates of change increase unboundedly with respect totime.

We also give the estimated order of convergence of the numerical solution to the

exact solution for each of the examples.

 Example 9.1

Let c ≡ 0. We consider the non-degenerate problem

ut  = uxx + 1(1 − u)θ 

, 0 < x < a, 0 < t < T , (9.63)

u(x, 0) = 0, 0 < x < a; u(0, t) = u(a,t) = 0, 0 < t < T , θ > 0 . (9.64)

We consider cases when θ  = 1/2, 1, and 2, respectively. According to investigations

by Acker, Walter, and Kawarada [1, 17, 32], the critical length a∗ ≈ 1.5303 for

θ  = 1. Our computations further indicate that a∗ ≈ 1.8856, 1.1832 for θ  =1/2 and 2, respectively. Let a = 1.55, 2, π, 10, respectively. We compute the

quenching time by means of the adaptive scheme developed. In the case of θ 

=1, a = π , Chan and Chen [4] show that 0.5 ≤ T a ≤ 0.6772. As for θ  = 1 anda = 1.55, 2, they later observe that T a = 3.963, 0.779, respectively. Let τ 0 =0.5 × 10−4 and h = 0.1 × 2−s , s = 0, 1, 2, 3, 4, respectively. Further, we let T M 

a be

the estimated quenching time obtained by several authors [2]–[5] via Crank–Nicolson

type schemes and Newton iterations, and denote ∞ as the case where no possibility of 

finite quenching time is detected. In Table 9.1(a), for each of the a > a∗ given, we list

the computed quenching time T a by using our compound adaptive scheme. We only

need to consider values of u at x = a/2 where maxima of the function u, 0 < x < a,

occur. Our results are almost identical to existing results but slightly less than those atthird decimal places [1], [4]–[7]. Numerical solutions also demonstrate good stability

of the scheme.

In Figures 9.1(b)–9.1(d), we plot evolution profiles of the numerical solution u,

as well as rates of change ut , ut t  for different testing values of a at x = a/2. The

monotone increases of the function values when a ≥ a∗ again demonstrate the con-

clusions of the lemma and theorem. It is also noticed that values of u, ut , and ut t 

increase smoothly at the beginning, but ut , ut t  grow exponentially while t approaches

T a . The phenomenon not only suggests the necessity of the use of finer time step

sizes through proper adaptive mechanisms near the quenching point, but also implies

that extra care is needed when designing or using a higher-order numerical method

for problems possessing quenching singularities (see Table 9.1(d) for maximal values

of functions u, ut , ut t ). The rapid increase of higher derivative values may enlarge

the error constants and may subsequently reduce the actual accuracy of a numeri-

cal method no matter how “higher order” is declared through a standard theoretical

analysis.

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Table 9.1 (a) The Computed Quenching Time T a for Different

θ  and a (τ 0 = 0.5 × 10−4)θ a \ r 0.1 0.4 1.6 6.4 25.6 T a T M 

a

0.5 1.55 ∞ ∞ ∞ ∞ ∞ ∞ N.A.

0.5 2 2.107 2.127 2.132 2.133 2.133 2.133 N.A.

0.5 π 0.776 0.774 0.774 0.773 0.773 0.773 N.A.

0.5 10 0.666 0.666 0.666 0.666 0.666 0.666 N.A.

1.0 1.55 3.669 3.893 3.942 3.957 3.961 3.961 3.963

1.0 2 0.778 0.778 0.779 0.779 0.779 0.779 0.779

1.0 π 0.539 0.539 0.538 0.537 0.537 0.537 0.5381.0 10 0.5 0.5 0.5 0.5 0.5 0.5 0.5

2.0 1.55 0.531 0.532 0.532 0.532 0.532 0.532 N.A.

2.0 2 0.401 0.400 0.400 0.400 0.400 0.400 N.A.

2.0 π 0.343 0.342 0.341 0.341 0.341 0.341 N.A.

2.0 10 0.333 0.333 0.333 0.333 0.333 0.333 N.A.

Table 9.1 (b) Computed Maximal Values of u, ut , ut t  Before Quench (θ =

1)

Max. values \ a 1.50 1.55 2.00 π 10.0

Max{u} 0.46 0.99 0.99 0.99 0.99

Max{ut } 1.02703 25.2960 35.3781 23.7981 40.4025

Max{ut t } 1.03107 12092.46 18254.49 15645.45 21013.38

Table 9.1 (c) The Monotone Convergence of T a as

a→ ∞a T a a T a a T a a.T a

1.55 3.961 1.80 0.999 3.00 0.546 5.00 0.503

1.60 2.007 1.90 0.871 π 0.537 10.00 0.500

1.70 1.257 2.00 0.779 4.00 0.511 50.00 0.500

Table 9.1 (d) The Approximation of α and β (θ  = 1; h, r = 0.1; a = 2,

τ 0 =

0.5×

10−

4)

t u(1, t) α β t u(1, t) α β

0.7780 0.96359 0.53622 1.47869 0.7785 0.97518 0.57526 1.96670

0.7781 0.96559 0.54252 1.54548 0.7786 0.97817 0.75067 7.75870

0.7782 0.96772 0.55606 1.70212 0.7787 0.98241 0.84052 16.0813

0.7783 0.97003 0.55561 1.69658 0.7788 0.98749 2.08430 641212.14

0.7784 0.97249 0.56439 1.81075 0.7789 0.99705 – –

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FIGURE 9.1

(a) The blow-up profile of the function ut  corresponding to the solution of (9.63)

and (9.64). Semi-adaptive method is used (θ  = 1; h = 0.1; τ 0 = 0.5 × 10−4

, a =1.55). It is noticed that the increase of ut   becomes exponential when t  ≥ 3 and

the solution reaches the quenching point at t  ≈ 3.669. ut   grows relatively slow

while t << 3.

FIGURE 9.1

(b) The profile of u under the same conditions of Figure 9.1(a). Semi-adaptive

method is used. It is noticed that the increase of u increases rapidly while t 

approaches the quenching time. u grows relatively slow while t << 3.

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Let a = 2. According to Filippas and Guo [11], the quenching solution satisfies

the following property:

limt →T a u(1, t) = 1 − √ 2(T a − t)1/2

, θ  = 1 . (9.65)

We wish to see if our numerical solution approximately satisfies (9.65). For this,

similar to Sanz–Serna et al. [30], we introduce an approximation of the formula (9.65)

with two parameters α, β:

u(1, t) = 1 − β(T a − t)α, θ  = 1 , (9.66)

where t  is sufficiently close to T a . By using the results in Table 9.1(a) for (9.66), we

immediately obtain Table 9.1(d) with estimated values of parameters α and β. Noticethe rapid growth of the numerical error as t  approaches T a  due to the quenching

singularity and the sensitivity of the formula (9.66) when used in calculations. Taking

the averages of their first six values from the table, we immediately obtain α =0.55507, β = 1.70005 which are good approximations to theoretical predictions.

Let u be the exact solution or its best known approximation, and uh,τ   the numerical

solutionof theproblem (9.63)and (9.64). Supposingthat theerror in thetime direction

is negligible, we may denote uh = uh,τ   and have |uh − u| ≈ Chρ . Let the profile of 

τ  be the same while h is reduced. The order of accuracy, ρ, can then be estimated by

means of the formula

ρ ≈ 1

ln 2ln

|uh − u||uh/2 − u| . (9.67)

To apply the formula, we set h = 0.1 and let u be the numerical solution at h =1/160 in the uniformed spatial interval. We only consider the case of θ  = 1, a = π

as an example here and leave the more general discussion to the nonlinear degenerate

problem in next example. Let τ 0

=0.5

×10−5. We take the initial time step τ  as

0.0001 so that the influence of error may be neglected. We compute the value of ρ at(a/2, t) for 0.438 ≤ t  = mτ < T a ≈ 0.538 and then consider the arithmetic average

as the estimate of the order of accuracy. We obtain

ρ ≈ 2.27557286

which fairly indicates that the actual order of accuracy is around 2. A similar estimate

can be computed in the time direction by choosing h2 << τ  and adopting an analog

of (9.67).

Finally,  Figure 9.1(e) shows the profile of the adaptive temporal discretizationparameter τ , while θ  = 1, a = π , initial τ  = 0.001 and τ 0 = 0.0.5 × 10−4 are

considered. Profiles of τ  in other cases are similar.

 Example 9.2

Let N  = 79 for the normalized interval [0, 1]. We consider the initial spatial step

size h = 1/80 in the space, while the initial temporal step being τ 0 = 0.001 in our

experiments.

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FIGURE 9.1

(e) The profile of τ . It is observed that the temporal step size decreases accordingto the monotone increase of ut  until the given minimal stepsize controller.

Consider the following degenerate semilinear initial-boundary value problem:

xq ut  = uxx + cux + 1

(1 − u)θ , 0 < x < r, 0 < t < T , q ≥ 0, θ > 0 , (9.68)

u(x, 0) = 0, 0 < x < r; u(0, t) = u(r, t) = 0, 0 ≤ t < T . (9.69)

We need only consider the case with θ  = 1 and other cases are similar. The function cis taken to be b/(1+x) and b/x, respectively, where b is a constant. We note that in the

latter case, the function c becomes unbounded while x → 0+ and this causes slight

perturbation at the left end of the interval [0, 1] in the numerical solution. However,

the amplitude of this oscillation decreases and is under control as the computation

continuous.

For the standard quenching problem (9.68) and (9.69), we wish to predict the

critical length rq,c  and quenching time T q,c  while computing the numerical solution,

should they exist and be finite.Figures 9.2(a) and  (b) show profiles of the derivative function ut   in during the

final stages before blowing up. The parameter q = 1 and interval length r = π

are used. Functions at 10 different time levels from 0.7434 to 0.7443 are displayed.

The maximal value of the derivative function ut   increases monotonically but rapidly

from 18.0961 when t  = 0.7434 to 82.5939 as t  reaches 0.7443. The location of the

maximal value in the space is approximately 1.2570 which is slightly shifted to the

left from the center of the interval.

In Table 9.2(a), we list locations of maxx ut (x,t) immediately before quenching:

The predicted quenching time in the case is T q,c = 0.7443+. In Figure 9.2(c), we

show the profile of the solution u during the same stage before quenching. Values of 

maxx u(x,t) tend to the unity monotonically and steadily. We have maxx,t ≤0.7443 =0.991742 in Figure 9.2(c). The same set of parameters as in previous graphs is used.

To see more precisely the shape of solution u, we darken the area under u, 0 < x < π .

We note that since that the coefficient function used, c(x) = −2/(1 + x), is bounded

throughout the computation, both the solution u and its derivative are smooth until

quench is reached.

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00.5

11.5

22.5

33.5

0.7434

0.7436

0.7438

0.744

0.7442

0

10

20

30

40

50

60

70

80

90

xt

     u      t

FIGURE 9.2

(a) The profile of the derivative function ut  . Because of the fully adaptive mesh

structure, here we plot only the numerical solution at 10 selected time levels

immediately before the quenching time (q = 1, c = −2/(1 + x),a = π).

0 0.5 1 1.5 2 2.5 3 3.5

0

10

20

30

40

50

60

70

80

90

x

     u      t

FIGURE 9.2

(b) The blow-up of the derivative function ut   (q = 1; c = −2/(1 + x),a = π).

Numerical solutions at the same 10 time levels as before are considered. We

observe that the amplitude of ut  changes rapidly as t  approaches the quenching

time.

Figure 9.2(d) is devoted to the distribution of the spatial step sizes immediately

before quench occurs. In the first picture, we plot the diagram as the lengths of 

80 spatial grids used. The height of each vertical bar is given by the step size hi , 1 ≤i ≤ 80. We find that the step size reduces rapidly when it gets closer to the quenching

location. The step sizes resume slightly at the predicted quenching location due to

the influence of the cubic spline approximation used in between different time levels.

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0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 9.2

(c) The solution u at the final time stage (q = 1; c = −2/(1 + x),a = π, t  =0.74412). We purposely darken the area topped by the solution in order to view

better the curve.

Table 9.2  (a) Maximal Values of ut (x,t) and Their Locations

t xmax maxx ut  t xmax maxx ut  t xmax maxx ut 

0.7434 1.257 18.0961 0.7438 1.257 23.6767 0.7442 1.257 46.6237

0.7436 1.257 20.1446 0.7440 1.257 30.0193 0.7443 1.257 82.5939

Sharp increases of the function can be observed during the final stage of computations

(q = 1,c(x) = b/(1 + x),b = −2, r = π ).

The second graph in Figure 9.2(d) displays the same distribution by plotting grid

references (xi , hi+1), i = 0, 1, . . . , 80, in the same map. We may again observe that

mesh points are well distributed according to the profile of ut , which is exhibited in

Figures 9.2(a) and (b).

Table 9.2 (b) Maximal Values of ut (x,t) and Their Locations

t xmax maxx ut  t xmax maxx ut  t xmax maxx ut 

0.5720 1.335 21.2765 0.5724 1.335 28.5837 0.5728 1.335 64.3619

0.5722 1.335 24.0991 0.5726 1.335 37.0148 0.5729 1.335 280.586

Sharp increases of the function can be observed during the final stage of computations

(q = 0.2,c(x) = b/x,b = 0.4, r = π).

Table 9.2(b) is for numerical experiments when a different coefficient function,

c(x) = 0.4/x, together with q = 0.2, r = π is employed. Again, the derivative

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0 10 20 30 40 50 60 70 800

0.02

0.04

0.06

0.08

0.1

0.12

n

       h     n     +       1

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

xn

       h     n     +

       1

FIGURE 9.2

(d) The grid distribution in space at t  = 0.74421, immediately before quenching.

The first diagram is for the distribution against n while the second one is against

the location x.

Table 9.2  (c) The Evolution of the Error Ratio Function ek/ek−1 in a Quenching

Case (q = 1,c(x) = b/(1 + x),b = −2, r = 5)

t k ek/ek−1 t k ek/ek−1 t k ek /ek−1 t k ek/ek−1

0.209500 0.999104 0.500509 0.996062 0.710001 1.00061 0.739001 1.00334

0.300500 0.999104 0.600017 0.996971 0.720001 1.00071 0.739101 1.00335

0.400038 0.999937 0.700001 1.000460 0.730001 1.00104 0.739201 243.771

It is observed, as predicted, the ratio blows up as t  → T q,c .

function ut   in the final stage is displayed. The function is plotted in 10 different

time levels with the stage as t  varies from 0.5578 to 0.5587. The maximal value,

maxx ut (x,t), tends to infinite as t  approaches T q,c  which is approximately 0.5729+in the case. The computed maximal value of the derivative function ut   immediately

before quenching is about 280.586.

In Table 9.2(c), we present values of the error ratio ek/ek−1, k = 1, . . . , K , for

a quenching case with c(x) = −2/(1 + x). Parameters q = 1, r = 5 are used.

Predicted quenching time T q,c ≈ 0.739201+. We observe that the ratio remains to be

well bounded before the quenching time and increases abruptly when quenching time

is reached. This provides an obvious stopping criterion as we discussed earlier in the

last section. Figure 9.2(e) further illustrates such stopping criterion. In the graph, we

plot out the error ratio ek/ek−1 while thecomputational advice goes by. Theparticular

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.810

1

100

10

1

102

103

tk

(quenching)

     e       k       /     e

       k       1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.0910

1

100

101

tk

(no quenching)

     e       k

       /     e       k

       1

FIGURE 9.2

(e) Error ratio profiles. Though initially the error ratio function ek /ek−1 looks

flat, it increases rapidly while t  approaches the quenching time, if quenching

occurs. Settings q = 1, b = −2/(1 + x),a = 5 (top), and a = 0.5 (bottom) are

used.

cases, quenching and non-quenching, are considered. For the quenching case (the

first graph), this ratio remains nearly as constant 1 and does not change much until

quenching occurs. At that point, it suddenly jumps from 1.00335 to 243.771 and

this indicates the break down of the numerical computation. The same ratio remains

steady in the second graph for the non-quenching case.From the above experiments, we may conclude that:

1. The linearly implicit semi- and fully adaptive schemes are highly efficient and

accurate for solving the nonlinear reaction-diffusion problems with singular

source terms. The systems of equations derived with nonsingular tridiagonal

coefficient matrices are relatively simple to solve and numerically stable. The

adaptive designs work smoothly throughout the computation.

2. The adaptive methods developed are also reliable. In the fully adaptive method,we not only employ adaptive structures both in space and time, but also build

a stopping criterion based on the error analysis. The numerical solution well

follows the pattern of the physical solution. With the help of the adaptation in

space, the break up of ut  in spatial directions can be more precisely monitored.

This is in fact difficult to achieve for time-only adaptive algorithms.

3. The adaptive structures discussed can be conveniently extended for solving

multidimensional singular reaction-diffusion problems defined in, say, rectan-

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gular spatial domains. We may predict that method of dimensional splitting

can be employed to further simplify the computational procedure. In that case,

the critical length will be replaced by the critical index which combines both

the physical size information and geometric pattern factor of the spatial domaininvolved.

Acknowledgments

The work of the first author is supported in part by the Louisiana State under theGrant No. LEQSF-(1997-00)-RD-B-15.

References

[1] A. Acker and W. Walter, The quenching problem for nonlinear parabolic dif-

ferential equations, Lecture Notes in Math., 564 (1976), 1–12, Springer-Verlag,

New York.

[2] M.C. Branch, M.W. Beckstead, T.A. Litzinger, M.D. Smooke, and V.H. Yang,

Nonsteady combustion mechanisms of advanced solid propellants, Annual

Technical Report, 94-05, Center for Combustion and Environmental Research,

University of Colorado, Boulder, CO., 1994.

[3] C.J. Budd, G.P. Koomullil, and A.M. Stuart, On the solution of convection-

diffusion boundary value problems using equidistributed grids, SIAM J. Sci.

Comput., 20, (1998), 591–618.

[4] C.Y. Chan and C.S. Chen, A numericalmethod for semilinear singular parabolic

mixed boundary-value problems, Quart. Appl. Math., 47, (1989), 45–57.

[5] C.Y. Chan, L. Ke, and A.S. Vatsala, Impulsive quenching for reaction-diffusion

equations, Nonlinear Anal., 22, (1994), 1323–1328.

[6] C.S. Chen, The method of fundamental solutions for nonlinear thermal explo-

sions, Comm. Numer. Methods Engrg., 11, (1995), 675–681.

[7] K. Deng and H.A. Levine, On the blow-up of ut  at quenching, Proc. Amer.

 Math. Soc., 106, (1989), 1049–1056.

[8] E.A. Dorfi and L.O.C. Drury. Simple adaptive grids for 1-D initial value prob-

lems, J. Comput. Phys., 40, (1981), 202.

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[9] E.S. Fraga and J.Ll. Morris, An adaptive mesh refinement method for nonlinear

dispersive wave equations, J. Comp. Physics, 101, (1992), 94–103.

[10] R.M. Furzeland, J.G. Verwer, and P.A. Zegeling, A numerical study of threemoving-grid methods for one-dimensional partial differential equations which

are based on the method of lines, J. Comp. Physics, 89, (1990), 349–388.

[11] S. Filippas and J.S. Guo, Quenching profiles for one-dimensional semilinear

heat equations, Quart. Appl. Math., 51, (1993), 713–729.

[12] A. Ghafourian, C. Huyn, P. Johnson, S. Hevert, H. Dindi, S. Mahalingam,

and J.W. Faily, Liquid rocket combustion instability, Research Report, 90-02,

(1990), Center for Combustion and Environmental Research, University of 

Colorado, Boulder, CO.

[13] J.S. Guo and B. Hu, The profile near quenching time for the solution of a

singular semilinear heat equation, Proc. Edinburgh Math. Soc., (2) 40, (1997),

437–456.

[14] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John

Wiley & Sons, Inc, New York, 1962.

[15] W. Huang, Y. Ren, and R.D. Russell, Moving mesh partial differential equations(MM-PDEs) based on the equidistribution principle, SIAM J. Numer. Anal., 31,

(1994), 709–730.

[16] A. Iserles, A First Course in the Numerical Analysis of Differential Equations,

Cambridge University Press, 1996.

[17] H. Kawarada, On solutions of initial-boundary value problem for ut  = uxx +1/(1 − u), Publ. Res. Inst. Math. Sci., 10, (1975), 729–736.

[18] A.Q.M. Khaliq, E.H. Twizell, and D.A. Voss, On parallel algorithms forsemidiscretized parabolic partial differential equations based on subdiagonal

padé approximations, Num. Math. for Partial Diff. Eqns., 9, (1993), 107–116.

[19] J. Lang, Two-dimensional fully adaptive solutions of reaction-diffusion equa-

tions, Appl. Numer. Math., 18, (1995), 223–240.

[20] J. Lang and A. Walter, An adaptive Rothe method for nonlinear reaction-

diffusion systems, Appl. Numer. Math., 13, (1993), 135–146.

[21] H.A. Levine, Quenching, nonquenching, and beyond quenching for solutionsof some parabolic equations, Ann. Mat. Pure. Appl., 4, (1989), 243–260.

[22] V. Pareyra and E.G. Sewell, Mesh selection for discrete solution of boundary

value problems in ordinary differential equations, Numer. Math., 23, (1975)

261–268.

[23] Y. Ren and R.D. Russell, Moving mesh techniques based upon equidistribution,

and their stability, SIAM J. Sci. Stat. Comput., 13, (1992) 1265–1286.

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[24] R.D. Russell and J. Christiansen, Adaptive mesh selection strategies for solving

boundary value problems, SIAM J. Numer. Anal., 15, (1978) 59–80.

[25] L.F. Shampine, Numerical Solution of Ordinary Differential Equations, Chap-man & Hall, 1994.

[26] Q. Sheng, A monotonically convergent adaptive method for nonlinear combus-

tion problems, Integral Methods in Science & Engineering, Research Notes in

Mathematics 418, Chapman & Hall/CRC (2000), 310–315.

[27] Q. Sheng and H. Cheng, A moving mesh approach to the numerical solution

of nonlinear degenerate quenching problems, Dynamic Sys. Appl., 7, (1999),

343–358.

[28] Q. Sheng and H. Cheng, An adaptive grid method for degenerate semilinear

quenching problems, Computers Math. Applications, 39, (2000), 57–71.

[29] Q. Sheng and A.Q.M. Khaliq, A compound adaptive approach to degenerate

nonlinear quenching problems, Numer. Meth. for PDEs, 15, (1999), 29–47.

[30] Y. Tourigny and J.M. Sanz-Serna, The numerical study of blow-up with ap-

plication to a nonlinear Schrödinger equation, J. Comp. Phys., 102, (1992),

407–416.

[31] D.A. Voss and A.Q.M. Khaliq, Parallel LOD methods for second order time

dependent PDEs, Computers Math. Applic., 30, (1995), 25–35.

[32] W. Walter, Parabolic differentialequationswith a singular nonlinear term, Funk-

cial Ekvac., 19, 271–277.

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Chapter 10

 Adaptive Linearly Implicit Methods for Heat and Mass Transfer Problems

J. Lang and B. Erdmann

10.1 Introduction

Dynamical process simulation is the central tool nowadays to assess the modeling

process for large-scale physical problems arising in such fields as biology, chemistry,

metallurgy, medicine, and environmental science. Moreover, successful numerical

methods are very attractive to design and control economical plants at low costs in a

short time. Due to the great complexity of the established models, the development

of fast and reliable algorithms has been a topic of continuing investigation during

recent years.

One of the important requirements that modernsoftware must meet today is to judge

the quality of its numerical approximations in order to assess safely the modeling

process. Adaptive methods have proven to work efficiently providing a posteriorierrorestimates andappropriate strategies to improve theaccuracy whereneeded. They

are now entering into real-life applications and starting to become a standard feature

in simulation programs. This chapter reports on one successful way to construct

discretization methods adaptive in space and time, which are applicable to a wide

range of practically relevant problems.

We concentrate on heat and mass transfer problems which can be written in the

form

B(x,t,u, ∇ u)∂t u = ∇ · (D(x,t,u, ∇ u)∇ u) + F(x,t,u, ∇ u) , (10.1)

supplemented with suitable boundary and initial conditions. The vector-valued solu-

tion u = (u1, . . . , um)T  is supposed to be unique. This problem class includes the

well-known reaction-diffusion equations and the Navier–Stokes equations as well.

In the classical method of lines (MOL) approach, the spatial discretization is done

once andkept fixed during the time integration. Discrete solution values correspond to

pointsonlines parallel to the timeaxis. Sinceadaptivity inspace means toadd ordelete

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points, in an adaptive MOL approach, new lines can arise and later disappear. Here,

we allow a local spatial refinement in each time step, which results in a discretization

sequence first in time then in space. The spatial discretization is considered as a

perturbation, which has to be controlled within each time step. Combined with a posteriori error estimates, this approach is known as adaptive Rothe method. First

theoretical investigations have been made by Bornemann [7] for linear parabolic

equations. Lang and Walter [26] generalized the adaptive Rothe approach to reaction-

diffusion systems. A rigorous analysis for nonlinear parabolic systems is given in

Lang [28]. For a comparative study, we refer to Deuflhard et al. [16].

Since differential operators give rise to infinite stiffness, often an implicit method

is applied to discretize in time. We use linearly implicit methods of Rosenbrock type,

which are constructed by incorporating the Jacobian directly into the formula. Thesemethods offer several advantages. They completely avoid the solution of nonlinear

equations, which means no Newton iteration has to be controlled. There is no problem

to construct Rosenbrock methods with optimum linear stability properties for stiff 

equations. According to their one-step nature, they allow a rapid change of step sizes

and an efficient adaptation of the spatial discretization in each time step. Moreover, a

simple embedding technique can be used to estimate the error in time satisfactorily.

A description of the main idea of linearly implicit methods is given in Section 10.2.

Stabilized finite elements are used for the spatial discretization to prevent numerical

instabilities caused by advection-dominated terms. To estimate the error in space, the

hierarchical basis technique has been extended to Rosenbrock schemes in Lang [28].

Hierarchical error estimators have been accepted to provide efficient and reliable

assessment of spatial errors. They can be used to steer a multilevel process, which

aims at getting a successively improved spatial discretization, drastically reducing

the size of the arising linear algebraic systems with respect to a prescribed tolerance

(Bornemann et al. [8], Deuflhard et al. [17], Bank and Smith [2]). A brief introduction

to multilevel finite element methods is given in Section 10.3.

The described algorithm has been coded in the fully adaptive software packageKardos at the Konrad–Zuse–Zentrum in Berlin. Several types of embedded Rosen-

brock solvers and adaptive finite elements were implemented. Kardos is based on

the Kaskade-toolbox [18], which is freely distributed at ftp://ftp.zib.de/pub/kaskade.

Nowadays both codes are efficient and reliable workhorses to solve a wide class of 

PDEs in one, two, or three space dimensions. To demonstrate the performance of our

adaptive approach, in Section 10.4 we will present two practically relevant problems

occurring in combustion theory and brine transport in porous media.

10.2 Linearly Implicit Methods

In this section a short description of the linearly implicit discretization idea is given.

More details can be found in the books of Hairer and Wanner [23], Deuflhard and

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Bornemann [15], and Strehmel and Weiner [37]. For ease of presentation, we first

set B = I  in (10.1) and consider the autonomous case. Then we can look at (10.1)

as an abstract Cauchy problem of the form

∂t u = f (u) , u(t  0) = u0 , t > t  0 , (10.2)

where the differential operators and the boundary conditions are incorporated into

the nonlinear function f(u). Since differential operators give rise to infinite stiffness,

often an implicit discretization method is applied to integrate in time. The simplest

scheme is the implicit (backward) Euler method

un+1 = un + τ f (un+1) , (10.3)

where τ  = t n+1 −t n is the step size and un denotes an approximation of u(t) at t  = t n.

This equation is implicit in un+1 and thus usually a Newton-like iteration method has

to be used to approximate the numerical solution itself. The implementation of an

efficient nonlinear solver is the main problem for a fully implicit method.

Investigatingtheconvergence ofNewton’s methodin functionspace, Deuflhard [13]

pointed out that one calculation of the Jacobian or an approximation of it per time

step is sufficient to integrate stiff problems efficiently. Using un as an initial iterate

in a Newton method applied to (10.3), we find

(I  − τ J n) Kn = τf(un) , (10.4)

un+1 = un + Kn , (10.5)

where J n stands for the Jacobian matrix ∂uf (un). The arising scheme is known as the

linearly implicit Euler method. The numerical solution is now effectively computed

by solving the system of linear equations that defines the increment Kn. Among

the methods that are capable of integrating stiff equations efficiently, linearly implicit

methods are the easiest to program, since theycompletely avoid the numericalsolution

of nonlinear systems.

One important class of higher-order linearly implicit methods consists of extrap-

olation methods that are very effective in reducing the error, see Deuflhard [14].

However, in the case of higher spatial dimension, several drawbacks of extrapolation

methods have shown up in numerical experiments made by Bornemann [6]. Another

generalization of the linearly implicit approach we will follow here leads to Rosen-

brock methods [35]. They have found wide-spread use in the ordinary differential

equations (ODE) context. Applied to (10.2) a so-called s-stage Rosenbrock method

has the recursive form

(I  − τ γ ii J n) Kni = τf un +

i−1j =1

αij  Knj 

+ τ J n

i−1j =1

γ ij  Knj  , i = 1(1)s , (10.6)

un+1 = un +

si=1

bi Kni , (10.7)

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where the step number s and the defining formula coefficients bi , αij , and γ ij  are

chosen to obtain a desired order of consistency and good stability properties for

stiff equations (see, e.g., [23, IV.7]). We assume γ ii = γ > 0 for all i, which is the

standard simplification to derive Rosenbrock methods with one and the same operatoron the left-hand side of (10.6). The linearly implicit Euler method mentioned above

is recovered for s = 1 and γ  = 1.

For the general system

B(t, u)∂t u = f (t , u) , u(t  0) = u0 , t > t  0 , (10.8)

an efficient implementation that avoids matrix-vector multiplications with the Jaco-

bian was given by Lubich and Roche [31]. In the case of a time- or solution-dependent

matrix B, an approximation of ∂t u has to be taken into account, leading to the gener-alized Rosenbrock method of the form

1

τ γ B(t n, un) − J n

U ni = f (t i , U i ) − B(t n, un)

i−1j =1

cij 

τ U nj  + τ γ i Cn

+ (B(t n, un) − B(t i , U i )) Zi , i = 1(1)s , (10.9)

where the internal values are given by

t i = t n + αi τ , U i = un +

i−1j =1

aij  U nj  , Zi = (1 − σ i )zn +

i−1j =1

sij 

τ U nj  ,

and the Jacobians are defined by

J n := ∂u(f(t,u) − B(t, u)z)|u=un,t =t n,z=zn ,

Cn := ∂t (f(t,u) − B(t, u)z)|u=un,t =t n,z=zn .

This yields the new solution

un+1 = un +

si=1

mi U ni

and an approximation of the temporal derivative ∂t u

zn+1 = zn +

si=1

mi

1τ 

ij =1

(cij  − sij )U nj  + (σ i − 1)zn

.

The new coefficients can be derived from αij , γ ij , and bi [31]. In the special case

B(t,u) = I , we get (10.6) setting U ni = τ 

j =1,...,i γ ij Knj , i = 1, . . . , s.

Various Rosenbrock solvers have been constructed to integrate systems of the

form (10.8). An important fact is that the formulation (10.8) includes problems

of higher differential index. Thus, the coefficients of the Rosenbrock methods have

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to be specially designed to obtain a certain order of convergence. Otherwise, order

reduction might happen. In [32, 31], the solver Rowdaind2 was presented, which is

suitable for semi-explicit index 2 problems. Among the Rosenbrock methods suitable

for index 1 problems we mention those given in [12, 29, 33, 36]. More informationcan be found in [28]. For the convenience of the reader, we give the defining formula

coefficients for Ros2 [12] and Rowdaind2 in Tables 10.1 and 10.2, respectively.

Both Rosenbrock solvers have been used in our simulations presented here.

Table 10.1 Set of Coefficients for Ros2 [12]

γ  = 1.707106781186547e + 00

a11 = 0.000000000000000e + 00 α1 = 0.000000000000000e + 00a21 = 5.857864376269050e − 01 α2 = 1.000000000000000e + 00a22 = 0.000000000000000e + 00

c11 = 5.857864376269050e − 01 s11 = 0.000000000000000e + 00c21 = 1.171572875253810e + 00 s21 = 3.431457505076198e − 01c22 = 5.857864376269050e − 01 s22 = 0.000000000000000e + 00

γ 1 = 1.707106781186547e + 00 σ 1 = 0.000000000000000e + 00γ 2 = −1.707106781186547e + 00 σ 2 = 5.857864376269050e − 01

m1 = 8.786796564403575e − 01 m1 = 5.857864376269050e − 01

m2 = 2.928932188134525e − 01 m2 = 0.000000000000000e + 00

Usually, one wishes to adapt the step size in order to control the temporal error.

For linearly implicit methods of Rosenbrock type, a second solution of inferior order,

say p, can be computed by a so-called embedded formula

un+1 = un +

s

i=1

mi U ni ,

zn+1 = zn +

si=1

mi

1

τ 

ij =1

(cij  − sij )U nj  + (σ i − 1)zn

,

where the original weights mi are simply replaced by mi . If p is the order of un+1,

we call such a pair of formulas to be of order p(p). Introducing an appropriate scaled

norm · , the local error estimator

rn+1 = un+1 − un+1 + τ (zn+1 − zn+1) (10.10)

can be used to propose a new time step by

τ n+1 =τ n

τ n−1

T OLt  rn

rn+1 rn+1

1/(p+1)

τ n . (10.11)

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Table 10.2 Set of Coefficients for Rowdaind2 [32, 31]

γ  = 3.000000000000000e − 01

a11 = 0.000000000000000e + 00 α1 = 0.000000000000000e + 00a21 = 1.666666666666667e + 00 α2 = 5.000000000000000e − 01a22 = 0.000000000000000e + 00 α3 = 1.000000000000000e + 00a31 = 1.830769230769234e + 00 α4 = 1.000000000000000e + 00a32 = 2.400000000000000e + 00a33 = 0.000000000000000e + 00a41 = 1.830769230769234e + 00a42 = 2.400000000000000e + 00a43 = 0.000000000000000e + 00

a44 = 0.000000000000000e + 00c11 = 3.333333333333333e + 00 s11 = 0.000000000000000e + 00c21 = 1.246438746438751e + 00 s21 = 5.555555555555556e + 00c22 = 3.333333333333333e + 00 s22 = 0.000000000000000e + 00c31 = −1.226780626780621e + 01 s31 = −4.239316239316217e + 00c32 = 4.266666666666667e + 01 s32 = 8.000000000000000e + 00c33 = 3.333333333333333e + 00 s33 = 0.000000000000000e + 00c41 = 5.824628046850726e − 02 s41 = −4.239316239316217e + 00c42 = 3.259259259259259e + 00 s42 = 8.000000000000000e + 00

c43 = −3.703703703703704e − 01 s43 = 0.000000000000000e + 00c44 = 3.333333333333333e + 00 s44 = 0.000000000000000e + 00

γ 1 = 3.000000000000000e − 01 σ 1 = 0.000000000000000e + 00γ 2 = 1.878205128205124e − 01 σ 2 = 1.666666666666667e + 00γ 3 = −1.000000000000000e + 00 σ 3 = 2.307692307692341e − 01γ 4 = 0.000000000000000e + 00 σ 4 = 2.307692307692341e − 01

m1 = 1.830769230769234e + 00 m1 = 2.214433650496747e + 00m2 = 2.400000000000000e + 00 m2 = 1.831186394371970e + 00m3 = 0.000000000000000e + 00 m3 = 8.264462809917363e − 03

m4 = 1.000000000000000e + 00 m4 = 0.000000000000000e + 00

Here, T OLt  is a desired tolerance prescribed by the user. This formula is related

to a discrete PI-controller first established in the pioneering works of Gustaffson et

al. [21, 20]. A more standard step-size selection strategy can be found in Hairer et

al. [22, II.4].

Rosenbrock methods offer several structural advantages. They preserve conserva-

tion properties like fully implicit methods. There is no problem to construct Rosen-

brock methods with optimum linear stability properties for stiff equations. Because

of their one-step nature, they allow a rapid change of step sizes and an efficient adap-

tation of the underlying spatial discretizations as will be seen in the next section.

Thus, they are attractive for solving real world problems.

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10.3 Multilevel Finite Elements

In the context of partial differential equations (PDEs), system (10.9) consists of 

linear elliptic boundary value problems possibly advection dominated. In the spirit

of spatial adaptivity, a multilevel finite element method is used to solve this system.

The main idea of the multilevel technique consists of replacing the solution space

by a sequence of discrete spaces with successively increasing dimension to improve

their approximation property. A posteriori error estimates provide the appropriate

framework to determine where a mesh refinement is necessary and where degrees

of freedom are no longer needed. Adaptive multilevel methods have proven to be

a useful tool for drastically reducing the size of the arising linear algebraic systemsand to achieve high and controlled accuracy of the spatial discretization (see, e.g.,

[1, 17, 27]).

Let T h be an admissible finite element mesh at t  = t n and S qh be the associated

finite dimensional space consisting of all continuous functions which are polynomials

of order q on each finite element T  ∈ T h. Then the standard Galerkin finite element

approximation U hni ∈ S qh of the intermediate values U ni satisfies the equation

Ln U 

hni , φ

= (rni , φ) for all φ ∈ S 

q

h , (10.12)

where Ln is the weak representation of the differential operator on the left-hand side

in (10.9) and rni stands for the entire right-hand side in (10.9). Since the operator Ln

is independent of i its calculation is required only once within each time step.

It is a well-known inconvenience that the solutions U hni may suffer from numerical

oscillationscaused by dominating convective and reactive terms as well. An attractive

way to overcome this drawback is to add locally weighted residuals to get a stabilized

discretization of the formLn U hni , φ

+

T ∈T h

Ln U hni ,w(φ)

T 

= (rni , φ) +

T ∈T h

(rni ,w(φ))T  , (10.13)

where w(φ) has to be defined with respect to the operator Ln (see, e.g., [19, 30, 38]).

Two important classes of stabilized methods are the streamline diffusion and the more

general Galerkin/least-squares finite element method.

The linear systems are solved by direct or iterative methods. While direct methods

work quite satisfactorily in one-dimensional and even two-dimensional applications,

iterative solvers such as Krylov subspace methods perform considerably better withrespect to CPU-time and memory requirements for large two- and three-dimensional

problems. We mainly use the Bicgstab-algorithm [40] with Ilu-preconditioning.

After computing the approximate intermediate values U hni a posteriori error esti-

mates can be used to give specific assessment of the error distribution. Considering

a hierarchical decomposition

S q+1h = S 

qh ⊕ Z

q+1h , (10.14)

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where Zq+1h is the subspace that corresponds to the span of all additional basis func-

tions needed to extend the space S qh to higher order, an attractive idea of an efficient

error estimation is to bound the spatial error by evaluating its components in the space

Zq+1h only. This technique is known as hierarchical error estimation and has been ac-

cepted to provide efficient and reliable assessment of spatial errors [8, 17, 2]. In [28],

the hierarchical basis technique has been carried over to time-dependent nonlinear

problems. Defining an a posteriori error estimator Ehn+1 ∈ Z

q+1h by

Ehn+1 = Eh

n0 +

si=1

mi Ehni , (10.15)

with Ehn0 approximating the projection error of the initial value un in Z

q+1h and Eh

ni

estimating the spatial error of the intermediate value U hni , the local spatial error for

a finite element T  ∈ T h can be estimated by ηT  := Ehn+1T . The error estimator

Ehn+1 is computed by linear systems which can be derived from (10.13). For practical

computations the spatially global calculation of Ehn+1 is normally approximated by

a small element-by-element calculation. This leads to an efficient algorithm for

computing a posteriori error estimates which can be used to determine an adaptive

strategy to improve the accuracy of the numerical approximation where needed. A

rigorous a posteriori error analysis for a Rosenbrock–Galerkin finite element method

applied to nonlinear parabolic systems is given in Lang [28]. In our applications we

applied linear finite elements and measured the spatial errors in the space of quadratic

functions.

In order to produce a nearly optimal mesh, those finite elements T  having an error

ηT  larger than a certain threshold are refined. After the refinement improved finite

element solutions U hni defined by (10.13) are computed. The whole procedure solve-

estimate-refine is applied several times until a prescribed spatial tolerance Ehn+1 ≤

T OLx is reached. To maintain the nesting property of the finite element subspaces,

coarsening takes place only after an accepted time step before starting the multilevel

process at a new time. Regions of small errors are identified by their η-values.

10.4 Applications

10.4.1 Stability of Flame Balls

The profound understanding of premixed gas flames near extinction or stability

limits is important for the design of efficient, clean-burning combustion engines and

for the assessment of fire and explosion hazards in oil refineries, mine shafts, etc.

Surprisingly, the near-limit behavior of very simple flames is still not well-known.

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Since these phenomena are influenced by buoyant convection, typically experiments

are performed in a µg environment. Under these conditions, transport mechanisms

such as radiation and small Lewis number effects, the ratio of thermal diffusivity

to the mass diffusivity, come into the play. Seemingly stable flame balls are oneof the most exciting appearances which were accidentally discovered in drop-tower

experiments by Ronney [34] and confirmed later in parabolic aircraft flights. First

theoretical investigations on purely diffusion-controlled stationary spherical flames

were done by Zeldovich [42]. Forty years later his flame balls were predicted to

be unstable [11]. However, encouraged by the above new experimental discoveries,

Buckmaster and collaborators [9] have shown  that for  low Lewis numbers flame

balls can be stabilized including radiant heat loss which was not considered before

(see  Figure 10.1 for a configuration of a stationary flame ball). Nowadays thereis an increasing interest in high-quality µg space experiments necessary to assess

the steady properties and stability limits of flame balls (see NASA information at

http://cpl.usc.edu/SOFBALL/sofball.html).

Fresh Mixture

Flame

Combustion Products

Heat and

(Reaction Zone)

Radiation

FIGURE 10.1

Configuration of a stationary flame ball. Diffusional fluxes of heat and combus-

tion products (outwards) and of fresh mixture (inwards) together with radiative

heat loss cause a zero mass-averaged velocity.

Although analytical modeling has identified the key physical ingredients of spheri-

calpremixed flames, quantitative confirmation canonly come from detailed numerical

simulations. Usually, spherically symmetric one-dimensional flame codes are used to

investigate steady properties, stability limits, and dynamics of flame balls (see, e.g.,

[10, 41]). Higher dimensional simulations are very rare due to their great demand

for local mesh adaptation in order to resolve the thin reaction layers. In [4] and [25]

two-dimensional computations of flame balls were presented. Three-dimensional

investigations using parallel architectures were published in [5].

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The mathematical model we shall adopt is that of [4], which is based on the constant

density assumption. In dimensionless form it reads

∂t T  − ∇ 2 T  = w − s ,

∂t Y  −1

Le∇ 2 Y  = −w ,

w =β2

2 LeY  exp

β(T  − 1)

1 + α(T  − 1)

, (10.16)

s = cT  

4− T  

4

u

(T b − T u)4.

Here, T  := (T  − T u)/(T b − T u) is the nondimensional temperature determined by

the dimensional temperatures T , T u, and T b, where the indices u and b refer to the

unburnt and burnt state of an adiabatic plane flame, respectively. Y  represents the

mass fraction of the deficient component of the mixture. The chemical reaction rate w

is modeled by a one-step Arrhenius term incorporating the dimensionless activation

energy β, the Lewis number Le, and the heat release parameter α := (T b − T u)/T b.

Heat loss is generated by a radiation term s modeled for the optically thin limit. The

strength of the radiative loss is mainly determined by the constant c, which depends

on the Stefan–Boltzmann constant and the Planck length. These relatively simpleequations are widely accepted to capture much of the essential physics of flame

balls [9, 10]. Comparisons of analytical treatments to experimental results provide

strong evidence of the model’s validity.

In the following computations, the conditions have been chosen similar to the

experiments made by Ronney [34] for a 6.5% H 2-air flame. We set T u = 300K,

T b = 830K, Le = 0.3, β = 10, and derive α = 0.64. In [34], additional CF 3Br as

a tracer concentration in the mixture was used to increase the heat loss by radiation.

Low concentration of CF 3Br yields cellular instability of the flame balls, whereas forincreased heat loss due to an increased concentration of the tracer, stable flame balls

can be observed. To simulate this behavior we use different values of c in (10.16),

here c = 0.01 and c = 0.1.

The computational domain has to be sufficiently large in order to avoid any dis-

turbance caused by the boundary. Typically, sizes of 100 times the flame ball radius

are needed to obtain domain-independent solutions due to the long far-field thermal

profiles [41]. In those cases, the conductive fluxes at the outer boundary are zero. We

consider domains = [−L, L]d , d  = 2, 3, with L = 200 according to the initial

flame radius r0 ∈ [0.2, 2.5]. As initial conditions we take the analytic solution for asteady plane flame in the high activation-energy limit [4, 5] and in some calculations

use a local stretching to generate an elliptic front. In the following we report on two

different scenarios, unstable and quasi-stationary flame balls.

Unstable two-dimensional flame balls. We set c = 0.01 and take an initial elliptic

flame with axis’ ratio of 1:4. After a short time an instability develops which results

in a local quenching of the flame as can be seen in Figures 10.2 and 10.3. After a

while the flame is split into two separate smaller flames, which separate again and

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FIGURE 10.2

Two-dimensional flame ball with Le = 0.3, c = 0.01. Iso-thermals T  =

0.1, 0.2, . . . , 1.0 at times t  = 10 and 30.

continue propagating. It can be seen that the dynamic spatial mesh chosen by our

adaptive algorithm for TOLt  = TOLx = 0.0025 is well-fitted to the behavior of the

solution. More grid points are automatically placed in regions of high activity inorder to resolve the steep solution gradients within the thin reaction layer.

Quasi-stationary two-dimensional flame balls. Fixing c = 0.1 in (10.16) and

varying the initial radii for a circular flame in a large number of calculations, we

found quasi-stationary flame ball configurations. In Figure 10.4 we have plotted the

evolution of the integrated reaction rate 

w(t, x, y) dx dy for selected initial radii.

For too small and too large radii, the flame is quickly extinguished. In between we

observe a convergence process to a quasi-steady state characterized by a very slow

decrease of the integrated reaction rate. The corresponding flame diameter is around

2. Similar results for c = 0.05 were reported in [4].

Splitting of three-dimensional flame balls. In the three-dimensional case we get a

more complex pattern formation. Just to give an impression, we select one typical

example taken from [5]. Starting with an ellipsoid having an axis ratio of 1:1:2, the

flame ball is split along the z-axis due to the thermo-diffusive instabilities and further

splitting occurs afterwards (see Figure 10.5). Although we were able to detect certain

parameter regions for extinction resulting from excessive heat loss, we have not yet

found configurations that are stable for longer time periods. This is the subject of 

current research.

10.4.2 Brine Transport in Porous Media

High-level radioactive waste is often disposed of in salt domes. The safety assess-

ment of such a repository requires the study of groundwater flow enriched with salt.

The observed salt concentration can be very high with respect to seawater, leading

to sharp and moving freshwater-saltwater fronts. In such a situation, the basic equa-

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FIGURE 10.3

Two-dimensional flame ball with Le = 0.3, c = 0.01. Reaction rate w at times

t  = 10, 30, and corresponding grids.

tions of groundwater flow and solute transport have to be modified [24]. We use the

physical model proposed by Trompert et al. [39] for a non-isothermal, single-phase,two-component saturated flow. It consists of the brine flow equation, the salt transport

equation, and the temperature equation, and reads

nρ (β ∂t p + γ ∂t w + α ∂t T ) + ∇ · (ρq) = 0 , (10.17)

nρ ∂t w + ρq · ∇ w + ∇ ·

ρJ w

= 0 , (10.18)

(ncρ + (1 − n)ρs cs )∂t T  + ρ c q · ∇ T  + ∇ · J T  = 0 , (10.19)

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0

5

10

15

20

25

0 5 10 15 20 25 30   I   N   T   E   G   R   A   L   O   F   R   E   A

   C   T   I   O   N

   R   A   T   E

TIME

r=0.2

r=0.3r=1.0

r=2.0

r=2.5

FIGURE 10.4

Two-dimensional flame ball with Le = 0.3, c = 0.1. Integrated reaction rate fordifferent initial radii.

FIGURE 10.5

Three-dimensional flame ball with Le = 0.3, c = 0.1. Iso-thermals T  = 0.8 at

times t  = 0.0, 2.0, 5.0, and 8.0.

supplemented with the state equations for the density ρ and the viscosity µ of the

fluid

ρ = ρ0 exp (α(T  − T 0) + β(p − p0) + γ w) ,

µ = µ0 (1.0 + 1.85w − 4.0w2) .

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Here, the pressure p, the salt mass fraction w, and the temperature T  are the inde-

pendent variables, which form a coupled system of nonlinear parabolic equations.

The Darcy velocity q of the fluid is defined as

q = −K

µ(∇ p − ρg) ,

where K is the permeability tensor of the porous medium, which is supposed to be of 

the form K = diag(k), and g is the acceleration of gravity vector. The salt dispersion

flux vector J w and the heat flux vector J T   are defined as

J w = −(nd m + αT |q|) I  +αL − αT 

|q|

qqT 

∇ w ,

J T  = −

(κ + λT |q|) I  +

λL − λT 

|q|qqT 

∇ T ,

where |q| = 

qT  q.

Writing the system of the three balance equations (10.17) through (10.19) in the

form (10.8), we find for the 3 × 3 matrix B

B( p, w, T) = nρβ nργ nρα

0 nρ 00 0 ncρ + (1 − n)ρs cs

.

Since the compressibility coefficient β is very small, the matrix B is nearly singular

and, as known [23, VI.6], linearly implicit time integrators suitable for differential

algebraic systems of index 1 do not give precise results. This is mainly due to the fact

that for β = 0 the matrix B becomes singular and additional consistency conditions

have to be satisfied to avoid order reduction. We have applied the Rosenbrock solver

Rowdaind2 [31], which handles both situations, β = 0 and β = 0.

An additional feature of the model is that the salt transport equation (10.18) isusually dominated by the advection term. In practice, global Peclet numbers can

range between 102 and 104, as reported in [39]. On the other hand, the temperature

and the flow equation are of standard parabolic type with convection terms of moderate

size.

Two-dimensional warm brine injection. This problem was taken from [39]. We

consider a (very) thin vertical column filled with a porous medium. This justifies the

use of a two-dimensional flow domain = {(x,y) : 0 < x, y < 1} representing a

vertical cross-section. The acceleration of gravity vector points downward and takesthe form g = (0, −g)T  , where the gravity constant g is set to 9.81. The initial values

at t  = 0 are

p(x,y, 0) = p0 + (1 − y)ρ0g, w(x , y , 0) = 0 , and T( x , y , 0) = T 0 .

The boundary conditions are described in Figure 10.6. We set wb = 0.25, T b =

292.0, and qb = 10−4. The remaining parameters used in the model are given in

Table 10.3.

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FIGURE 10.6

Two-dimensional brine transport. Computational domain and boundary con-

ditions for t > 0. The two gates where warm brine is injected are located at

(x,y) : 111

≤ x ≤ 211

, 911

≤ x ≤ 1011

, y = 0.

Table 10.3 Parameters of the Two-Dimensional Brine Transport

Model

n Porosity 0.4k Permeability 10−10 m2

d m Molecular diffusion 0.0 m2s−1

αT  Transversal dispersivity 0.002 mαL Longitudinal dispersivity 0.01 m

c Heat capacity 4182 J kg−1K−1

cs Solid heat capacity 840 J kg−1K−1

κ Heat conductivity 4.0 J s−1m−1K−1

λT  Transversal heat conductivity 0.001 J m−2K−1

λL Longitudinal heat conductivity 0.01 J m−2K−1

ρs Solid density 2500 kg m−3

ρ0 Freshwater density 1000 kg m−3

T 0 Reference temperature 290 K

p0 Reference pressure 105 kg m−1s−2

α Temperature coefficient −3.0 · 10−4 K−1

β Compressibility coefficient 4.45 · 10−10 m s2kg−1

γ  Salt coefficient ln(1.2)

µ0 Reference viscosity 10−3

kg m−1

s−1

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0.001

0.01

0.1

1

10

100

1000

0.001 0.01 0.1 1 10 100 1000 10000

   L   O   G   1   0   (   S   T   E   P

   S   I   Z   E   )

LOG10(TIME)

Step Size Control

0

500

1000

1500

2000

2500

30003500

4000

4500

5000

0.001 0.01 0.1 1 10 100 1000 10000

   N   U   M   B   E   R

   O   F   P   O   I   N   T   S

LOG10(TIME)

Degrees of Freedom

FIGURE 10.8

Two-dimensional brine transport. Evolution of time steps and number of spatial

discretization points for T OLt  = T OLx = 0.005.

values at t  = 0 are taken as

p(x,y,z, 0) = p0 + (0.03 − 0.012x + 1.0 − z)ρ0g, w(x, y, z, 0) = 0 ,

and the boundary conditions are

p = p(x,y,z, 0) , w = 0 , on x = 0 ,

p = p(x,y,z, 0) , ∂nw = 0 , on x = 2.5 ,

q2 = 0 , ∂nw = 0 , on y = 0 and y = 1 ,

q3 = 0 , ∂nw = 0 , on z = 0 and {z = 1} \ S ,

ρq3 = −0.0495, w = wb = 0.0935 , on S .

The parameters used in the three-dimensional simulation are given in Table 10.4.

Additionally, the state equation for the viscosity of the fluid is modified to

µ = µ0

1.0 + 1.85w − 4.1w

2

+ 44.5w

3.

In Figure 10.9 we show the distribution of the salt concentration in the plane y =

0.28125 after 2 and 4 h. The pollutant is slowly transported by the flow while

sinking to the bottom of the tank. The steepness of the solution is higher in the

back of the pollution front, which causes fine meshes in this region. Despite the

dominating convection terms, no wiggles are visible, especially at the inlet. An

interestingobservation is the unexpected drift in front of the solution— a phenomenon

which was also observed by Blom and Verwer [3].

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FIGURE 10.9

Three-dimensional brine transport. Level lines of the salt concentration w =

0.0, 0.01, . . . , 0.09, in the plane y = 0.28125 after 2 h (top) and 4 h (middle), and

corresponding spatial grids (bottom) in the neighborhood of the inlet.

10.5 Conclusion

Dynamical process simulation of complex real-life problems advises the use of 

modern algorithms, which are able to judge the quality of their numerical approx-

imations and to determine an adaptation strategy to improve their accuracy in both

the time and the space discretization. This chapter presented a combination of effi-

cient linearly implicit time integrators of Rosenbrock type and error-controlled grid

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Table 10.4 Parameters of the Three-Dimensional Brine

Transport Model

n = 0.35 γ  = ln(2) nd m = 10−9

κ = 7.18 · 10−11 αT  = 0.001 αL = 0.01p0 = 0.0 µ0 = 0.001 ρ0 = 1000

improvement based on a multilevel finite element method. This approach leads to a

minimization of the degrees of freedom necessary to reach a prescribed error toler-

ance. The savings in computing time are substantial and allow the solution of even

complex problems in a moderate range of time.

References

[1] R.E. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differen-

tial Equations — User’s Guide 8.0, SIAM, 1998.

[2] R.E. Bank and R.K. Smith, A Posteriori Error Estimates Based on Hierarchical

Bases, SIAM J. Numer. Anal., 30, (1993), 921–935.

[3] J.G. Blom and J.G. Verwer, VLUGR3: A Vectorizable Adaptive Grid Solver

for PDEs in 3D, I. Algorithmic Aspects and Applications, Appl. Numer. Math.,

16, (1994), 129–156.

[4] H. Bockhorn, J. Fröhlich, and K. Schneider, An Adaptive Two-Dimensional

Wavelet-Vaguellette Algorithm for the Computation of Flame Balls, Combust.

Theory Modeling, 3, (1999), 177–198.

[5] H. Bockhorn, J. Fröhlich, W. Gerlinger, and K. Schneider, Numerical Investiga-

tions on the Stability of Flame Balls, in: K. Papailiou et al., eds., Computational

Fluid Dynamics ’98, Vol. 1, John Wiley & Sons, New York, (1998), 990–995.

[6] F.A. Bornemann, An Adaptive Multilevel Approach to Parabolic Equations. II.

Variable-Order TimeDiscretization Based on a Multiplicative Error Correction,

 IMPACT of Comput. in Sci. and Engrg., 3, (1991), 93–122.

[7] F.A. Bornemann, An Adaptive Multilevel Approach to Parabolic Equations.

III. 2D Error Estimation and Multilevel Preconditioning, IMPACT of Comput.

in Sci. and Engrg., 4, (1992), 1–45.

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[8] F.A. Bornemann, B. Erdmann, and R. Kornhuber, A Posteriori Error Estimates

for Elliptic Problems in Two and Three Space Dimensions, SIAM J. Numer.

 Anal., 33, (1996), 1188–1204.

[9] J.D. Buckmaster, G. Joulin, and P.D. Ronney, Effects of Heat Loss on the

Structure and Stability of Flame Balls, Combust. Flame, 79, (1990), 381–392.

[10] J.D. Buckmaster, M. Smooke, and V. Giovangili, Analytical and Numerical

Modeling of Flame-Balls in Hydrogen-Air Mixtures, Combust. Flame, 94,

(1993), 113–124.

[11] J.D. Buckmaster and S. Weeratunga, The Stability and Structure of Flame-

Bubbles, Comb. Sci. Tech., 35, (1984), 287–296.

[12] K. Dekker and J.G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlin-

ear Differential Equations, North-Holland Elsevier Science Publishers, 1984.

[13] P. Deuflhard, Uniqueness Theorems for Stiff ODE Initial Value Problems, in:

D.F. Griffiths and G.A. Watson, eds., Numerical Analysis 1989, Proceedings

of the 13th Dundee Conference, Pitman Research Notes in Mathematics Series

228, Longman Scientific and Technical, (1990), 74–87.

[14] P. Deuflhard, Recent Progress in Extrapolation Methods for Ordinary Differ-

ential Equations, SIAM Rev., 27, (1985), 505–535.

[15] P. Deuflhard and F. Bornemann, Numerische Mathematik II, Integration

Gewöhnlicher Differentialgleichungen, De Gruyter Lehrbuch, Berlin, New

York, 1994.

[16] P. Deuflhard, J. Lang, and U. Nowak, Adaptive Algorithms in Dynamical Pro-

cess Simulation, in: H. Neunzert, ed., Progress in Industrial Mathematics at 

 ECMI’94, Wiley–Teubner, (1996), 122–137.

[17] P. Deuflhard, P. Leinen, and H. Yserentant, Concepts of an Adaptive Hierar-chical Finite Element Code, IMPACT of Comput. in Sci. and Engrg., 1, (1989),

3–35.

[18] B. Erdmann, J. Lang, and R. Roitzsch, KASKADE Manual, Version 2.0, Report

TR93-5, Konrad–Zuse–Zentrum für Informationstechnik, Berlin, 1993.

[19] L.P. FrancaandS.L. Frey, Stabilized FiniteElement Methods, Comput. Methods

 Appl. Mech. Engrg., 99, (1992), 209–233.

[20] K. Gustafsson, Control-Theoretic Techniques for Stepsize Selection in Implicit

Runge–Kutta Methods, ACM Trans. Math. Software, 20, (1994), 496–517.

[21] K. Gustafsson, M. Lundh, and G. Söderlind, A PI Stepsize Control for the

Numerical Solution of Ordinary Differential Equations. BIT 28, (1988), 270–

287.

[22] E. Hairer, S.P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations

 I, Nonstiff Problems, Springer-Verlag, Berlin, 1987.

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[23] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and 

 Differential-Algebraic Problems, Second Revised Edition, Springer-Verlag,

Berlin, 1996.

[24] S.M. Hassanizadeh and T. Leijnse, On the Modeling of Brine Transport in

Porous Media, Water Resources Research, 24, (1988), 321–330.

[25] L. Kagan and G. Sivashinski, Self-Fragmentation of Nonadiabatic Cellular

Flames, Combust. Flames, 108 (1997), 220–226.

[26] J. Lang and A. Walter, A Finite Element Method Adaptive in Space and Time

for Nonlinear Reaction-Diffusion Systems, IMPACT of Comput. in Sci. and 

 Engrg., 4, (1992), 269–314.

[27] J. Lang, Adaptive FEM for Reaction-Diffusion Equations, Appl. Numer. Math.,

26, (1998), 105–116.

[28] J. Lang, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems.

Theory, Algorithm, and Applications, Lecture Notes in Computational Science

and Engineering, Vol. 16, Springer-Verlag, Berlin, 2000.

[29] J. Lang and J. Verwer, ROS3P — an Accurate Third-Order Rosenbrock Solver

Designed for Parabolic Problems, Report MAS-R0013, CWI, Amsterdam, 2000.

[30] G. Lube and D. Weiss, Stabilized Finite Element Methods for Singularly Per-

turbed Parabolic Problems, Appl. Numer. Math., 17, (1995), 431–459.

[31] Ch. Lubich and M. Roche, Rosenbrock Methods for Differential-Algebraic

Systems with Solution-Dependent Singular Matrix Multiplying the Derivative,

Comput., 43, (1990), 325–342.

[32] M. Roche, Runge–Kutta and Rosenbrock Methods for Differential-Algebraic

Equations and Stiff ODEs, Ph.D. thesis, Université de Genève, 1988.

[33] M. Roche, Rosenbrock Methods for Differential Algebraic Equations, Numer.

 Math., 52, (1988), 45–63.

[34] P.D. Ronney, Near-Limit Flame Structures at Low Lewis Number, Combust.

Flame, 82, (1990), 1–14.

[35] H.H. Rosenbrock, Some General Implicit Processes for the Numerical Solution

of Differential Equations, Computer J., (1963), 329–331.

[36] G. Steinebach, Order-Reduction of ROW-methods for DAEs and Method of 

Lines Applications, Preprint 1741, Technische Hochschule Darmstadt, Ger-

many, 1995.

[37] K. Strehmel and R. Weiner, Linear-implizite Runge–Kutta–Methoden und ihre

Anwendungen, Teubner Texte zur Mathematik 127, Teubner Stuttgart, Leipzig,

1992.

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[38] L. Tobiska and R. Verfürth, Analysis of a Streamline Diffusion Finite Element

Method for the Stokes and Navier–Stokes Equation, SIAM J. Numer. Anal., 33,

(1996), 107–127.

[39] R.A. Trompert, J.G. Verwer, and J.G. Blom, Computing Brine Transport in

Porous Media with an Adaptive-Grid Method, Int. J. Numer. Meth. Fluids, 16,

(1993), 43–63.

[40] H.A. van der Vorst, BI-CGSTAB: A fast and smoothly converging variant of 

BI-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat., 13,

(1992), 631–644.

[41] M.S. Wu, P.D. Ronney, R.O. Colantonio, and D.M. Vanzandt, Detailed Numer-

ical Simulation of Flame Ball Structure and Dynamics, Combust. Flame, 116,(1999), 387–397.

[42] Ya.B. Zeldovich, Theory of Combustion and Detonation of Gases, Academy of 

Sciences (USSR), 1944.

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Chapter 11

Unstructured Adaptive Mesh MOL Solvers for Atmospheric Reacting-Flow Problems

M. Berzins, A.S. Tomlin, S. Ghorai, I. Ahmad, and J. Ware

11.1 Introduction

In this chapter, the method of lines (MOL) is applied to computational models of reacting flow arising from atmospheric applications. These computational models

describe the chemical transformations and transport of species in the troposphere

and have an essential role in understanding the complex processes which lead to

the formation of pollutants such as greenhouse gases, acid rain, and photochemical

oxidants. In order to make good comparisons with the limited experimental data

available, it is important to have a high degree of computational resolution, but at the

same time to model emissions from many different sources and over large physical

domains. This chapter is thus concerned with how to achieve this by using MOL

combined with spatial mesh adaptation techniques.

Achieving high resolution in air pollution models is a difficult challenge because of 

the large number of species present in the atmosphere. The number of chemical rate

equations that need to be solved rises with the number of species, and for high resolu-

tion 3-dimensional calculations, detailed chemical schemes can become prohibitively

large. The range of reaction time-scales often leads to stiff systems of differential

equations which require more expensive implicit numerical solvers. Previous work 

has shown [31, 32, 33, 12, 13] that coarse horizontal resolution can have the effect

of increasing horizontal diffusion to values many times greater than that describedby models, resulting in the smearing of pollutant profiles and an underestimation of 

maximum concentration levels. A review paper by Peters et al. [22] highlights the

importance of developing more efficient grid systems for the next generation of air

pollution models in order to “capture important smaller-scale atmospheric phenom-

ena.”

In general, the effects of mesh resolution have been well noted by the atmospheric

modeling community and attempts have been made to improve mesh resolution at

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the same time as trying to avoid excessive extra computational work. The usual

approach is to use nested or telescopic grids, where the mesh is refined in certain

regions of the horizontal domain which are considered of interest [15, 24, 30, 26].

This may include, for example, regions of high emissions such as urban areas, orclose to regions where significant monitoring is taking place. Previous work [32]

has shown, however, that such telescopic grids often cannot resolve plume structures

occurring outside of the nested regions and that adaptive refinement in the horizontal

domaincan provide higheraccuracy without entailing large extracomputational costs.

The primary reason is that away from concentrated sources, such models use large

grids of up to 50 km. Since dispersion can carry species distances of hundreds of 

kilometers from the source, such predescribed telescopic gridding models could still

lead to inaccurate downwind profiles as the plumes travel into those areas with largergrids. This is a particular problem when modeling species such as ozone, where the

chemical time-scale of pollutant formation is such that the main pollution episodes

occur at very long distances downwind of the sources of photochemical precursors.

The regions of steep spatial gradients of species such as ozone will move with time

according to thewind-fieldpresent and thespatial distribution of emissions. A reliable

solution can only be obtained if the mesh can be refined accordingly. The fine-scale

grids used in present regional scale models are of the order of 10 to 20 km. For a

power plant plume with a width of approximately 20 km, it is impossible to resolve

the fine structure within the plume using grids of this size. Furthermore, to refine themesh a priori, according to the path of the plume, would be an impossible task since

the plume position is a complicated function of many factors, including reaction,

deposition, and transport. There is a need for the application of methods which

can refine the grid according to where the solution requires it, i.e., time-dependent

adaptive algorithms. While there have been some applications of adaptive grids for

environmental modeling, e.g., Skamarock et al. [27], as yet these methods have not

been implemented in standard air quality models.

This chapter is based on the work done by the authors in applying adaptive grid-

ding techniques, which automatically refine the mesh in regions of high spatial error,

and illustrates the benefits this can bring over the telescopic approach in which mesh

refinement is only used close to a pollution source. The first part of this chapter

(Sections 11.2 to 11.4) describes the algorithms used and presents results for the 1D

hyperbolic conservation law with a nonlinear source term, of Leveque and Yee [18].

This deceptively simple problem may be used to show that spurious numerical so-

lution phenomena, such as incorrect wave speeds may occur when insufficient spa-

tial and temporal resolution are used. Sections 11.5 to 11.10 of the chapter willprovide a summary of the results for more complex two-dimensional atmospheric

problems (see [32]) while three-dimensional problems (see [33, 12]) are considered

in Sections 11.10 to 11.14. The general approach used here is to employ positivity-

preserving spatial discretization schemes in the method of lines to reduce the partial

differential equation (PDE) to a system of ordinary differential equations (ODEs) in

time. For reacting-flow problems, the numerical results will show that spatial mesh

points should be chosen with great care to reflect the true solution of the PDE and to

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avoidgeneratingsignificant but spurious numerical solution features. This is achieved

here by using adaptive mesh algorithms [3] to control the spatial discretization error

by refining and coarsening the mesh.

As reacting-flow problems require the use of implicit methods to resolve the fasttransients associated with some chemistry species, the cost of using implicit meth-

ods may be high unless great care is taken with numerical linear algebra. In the

present work this is done by making use of an approach developed for atmospheric

chemistry solvers [35, 2]. This approach uses a Gauss–Seidel iteration applied to the

source terms alone. The advective terms are effectively treated explicitly but without

introducing a splitting error. In three dimensions because of the need to preserve pos-

itivity of the solution and to be more concerned about efficiency, we have also used

a more traditional operator-splitting approach. In particular, the overall conclusionto be drawn from the computational evidence for one-, two-, and three-dimensional

problems is that having good mesh resolution in certain parts of the solution domain

is of critical importance with regard to obtaining a meaningful solution.

11.2 Spatial Discretization and Time IntegrationThe 1D Leveque and Yee problem [18] is given by

∂u

∂t +

∂u

∂x= −ψ (u) x ∈ [0, ∞], ψ (u) = µu(u − 1)(u − 0.5) (11.1)

and is linear advection with a source term that is “stiff” for large µ. The initial and

boundary values (at x = 0) are defined by

u(x, 0) = u0(x) = uL = 1, x ≤ xd ; uR = 0, x > xd 

where xd  = 0.1 or 0.3 in the cases considered here. The infinite domain will also be

truncated to [0, 1] for the cases considered here, as this is sufficient to demonstrate

the behavior of the methods employed. A simple outflow boundary condition is

then used at x = 1. The solution of Equation (11.1) is a discontinuity moving with

constant speed and has a potentially large source term that only becomes active at the

discontinuity [18].

Define a spatial mesh 0 = x1

< · · · < xN  = 1 and the vector of values U  with

components U i (t) ≈ u(xi , t) where u(x,t) is the exact solution to the PDE. We

define U i (t) as the exact solution to the ordinary differential equation (ODE) system

derived by spatial semi-discretization of the PDE and given by

U  = F N (t , U (t )), U (0) given . (11.2)

This true solution [U (t n)]pn=0 is approximated by [V (t n)]

pn=0 at a set of discrete times

0 = t 0 < t 1 < · · · < t p = t e by the time integrator. The form of the ODE system

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given by Equation (11.2) at time t  is given by

F N (t n, U ( t  n)) = F f N (t n, U ( t  n)) + F sN (t n, U ( t  n)) , (11.3)

where the superscripts f  and s denote the flow and source term parts of the func-

tion F  as defined below. The function F f N (t n, U ( t  n)) is the second-order limited

discretization of the advective terms in Equation (11.1) whose components are given

by

F f j  (t,U(t)) = −

1 +

(B(rj , 1)

2−

B(rj −1, 1)

2rj −1

(U j (t) − U j −1(t))

x. (11.4)

The function B is a limiter such as that of van Leer: (see [3])

B(rj , 1) =rj  + | rj  |

1 + rj 

, and rj  =U j +1(t) − U j (t)

U j (t) − U j −1(t). (11.5)

The vector F sN (t,U(t)) represents the approximate spatial integration of the source

term which is defined by 1x

 xj + 12

xj − 1

2

ψ(U(x,t))dx and is evaluated by using the mid-

point quadrature rule so that its j th component is:

F s

j (t,U j (t)) = ψ(U j (t)) . (11.6)

The time integration method used here (mostly for simplicity of analysis) is the

Backward Euler method defined by

V (t n+1) = V (t n) + F N (t n+1, V ( t  n+1)) . (11.7)

In the case when a modified Newton method is used to solve the nonlinear equations

at each timestep, this constitutes the major computational task of a method of lines

calculation. In cases where large ODE systems result from the discretization of flow

problems with many chemical species, the CPU times may be excessiveunless specialiterative methods are used.

The approach of [4] is used to neglect the advective terms J f  =∂F f 

∂V , and to con-

centrate on the Jacobian of the source terms J s =∂F s

∂V when forming the Newton

iteration matrix. This approach, in the case when no source terms are present, cor-

responds to using functional iteration for the advective calculation, see [2, 4]. The

matrix I  − tγ J s is the Newton iteration matrix of that part of the ODE system

corresponding to the discretization of the time derivatives and the source terms alone.

This matrix is thus block-diagonal with as many blocks as there are spatial elementsand with each block having as many rows and columns as there are PDEs. The

fact that a single block relates only to the chemistry within one cell means that each

block’s equations may be solved independently by using a Gauss–Seidel iteration.

This approach has been used with great success for atmospheric chemistry problems

[35]. The nonlinear equations iteration employed here may thus be written as

[I  − t J s ]

V m+1 (t n+1) − V m (t n+1)

= r

t mn+1

(11.8)

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where r

t mn+1

= −V m(t n+1) + V (t n) + tF N (t n+1, V m(t n+1)). Providing that the

iteration converges, this approximation has no adverse impact on accuracy. In order

for this iteration to converge with a rate of convergence rc it is necessary [2] that

|| [I  − tJ s ]−1 t J f  || = rc where rc < 1 . (11.9)

Using the identity ab ≤ a b , and defining J ∗f  as J ∗f  = (x)J f  gives:

t 

x|| J ∗f  || ≤ rc || [I  − tJ s ] || . (11.10)

Hence the convergence restriction may be interpreted as a CFL type condition. For

example, in the case of the PDE in (11.1), [I − tJ s

] is a diagonal matrix with entries

1 + tµ∂ψ∂V 

where

∂ψ

∂V = p(V) (11.11)

and where p(V) = 3V 2 − 3V  + 0.5 gives a CFL type condition that allows larger

timesteps as µ increases. The function p(V) is bounded between the values 0.5 and

−0.25 for solution values in the range [0, 1].

11.3 Space-Time Error Balancing Control

Hyperbolic PDEs are often solved by using a CFL condition to select the timestep.

The topic of choosing a stable stepsize for such problems has been considered in detail

by Berzins and Ware [6]. Although a CFL condition indicates when the underlying

flow without reactions is stable, it is still necessary to get the required accuracy for thechemistry terms. In most time dependent PDE codes, either a CFL stability control is

employed or a standard ODE solver is used which controls the local error ln+1(t n+1)

with respect to a user supplied accuracy tolerance. Efficient time integration requires

that the spatial and temporal errors are roughly the same order of magnitude. The

need for spatial error estimates unpolluted by temporal error requires that the spatial

error is the larger of the two. One alternative approach developed by Berzins [3, 4]

is to use a local error per unit step control in which the time local error (denoted by

le(t)) is controlled so as to be smaller than the local growth in the spatial error over thetimestep (denoted by est(t)). In the case of the Backward Euler method, the standard

local error estimate at t n+1 is defined as le(t n+1) and is estimated in standard ODE

codes by

le(t n+1) =t 

2

F N (t n+1, V ( t  n+1)) − F N (t n, V ( t  n))

.

≈t 2

2V (t n+1) (11.12)

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where the function F  is defined by Equation (11.2). The error control of [3] is defined

by

len+1(t n+1) ≤ est(t n+1) (11.13)

where 0 < < 1 is a balancing factor and est (t n+1) represents the local growth

in time of the spatial discretization error from t n to t n+1, assuming that the error

is zero at t n. Once the primary solution has been computed using the method of 

Section 11.2, a secondary solution is estimated at the same time step with an upwind

scheme of different order and a different quadrature rule for source-term integration.

The difference of these two computed solutions is then taken as an estimate of the

local growth in time of the spatial discretization error in the same way as in [3]. The

primary solution V (t n+1) starting from V (t n) is computed in the standard way asdescribed in Section 11.2. The secondary solution W (t n+1) is computed by solving

W(t) = Gf (t,W(t)) + Gs (t, W(t)), W (t n) = V (t n) . (11.14)

with initial value V n, whereGf  and Gs are the first-order advective term and the

source terms which are evaluated using a linear approximation on each interval and

the trapeziodal rule, i.e.,

Gf j  (t,W j (t)) = −

(W j (t) − W j −1(t)

x

Gsj (t,W j (t)) =

1

4(ψ(W j −1(t)) + 2ψ(W j (t)) + ψ(W j +1(t))) . (11.15)

Estimating est(t n+1) by applying the Backward Euler Method to (11.14) subtracted

from (11.7) withone iteration of the modified Newton iteration of the previous section,

as in [4], gives

[I  − tJ s ]

est (t n+1)

=t 

F f 

t n+1, V  (t n+1)

− Gf 

t n+1, V  (t n+1)

+ F s

t n+1, V  (t n+1)

− Gs

t n+1, V  (t n+1)

(11.16)

where est(t n+1) ≈ V (t n+1) − W (t n+1).

11.4 Fixed and Adaptive Mesh Solutions

In the case of the problem defined by Equation (11.1), comparisons were made

between the standard local error control approach in which absolute and relative

tolerances RTOL and ATOL are defined (see, [21]), and the new approach defined

by (11.13). The choice of the parameter is an important factor in the performance

of the second approach. In selecting this parameter the local growth in the spatial

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discretization error should dominate the temporal error and the work needed should

not be excessive. Obviously the larger the value of  the fewer ODE time steps there

willbe, and the smaller the valueof  the moresteps there will be. Agood compromise

between efficiency and accuracy is to use in the range of 0.1 to 0.3. The numericalexperiments described by Ahmad [1] confirm the results of Berzins [3], although it

is noted that for some combustion problems, may have to be reduced to below 0.1.

An important feature of solving the problem defined by Equation (11.1) is that the

numerical solution may move with an incorrect wave speed. Leveque and Yee [18]

showed that the stepsizeand the meshsize shouldbeO( 1µ

), toavoid spurious solutions

being generated. In order to illustrate these results we have taken xd  = 0.3 in

Equation (11.1), x = 0.02 and used a fixed time step t  = 0.015. The product of 

time step t and the reaction rate µ determines thestiffness of thesystem. Figure11.1shows the comparison of the computed solution and exact solution at t  = 0.3 for µ =

100, and 1000 (tµ = 1.5 and 15), respectively. It is evident from Figure 11.1

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

   S  o   l  u   t   i  o  n

x

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

   S  o   l  u   t   i  o  n

x

FIGURE 11.1

Comparison between true solution (line) and numerical solution (dots) using

local error control with 0.01 relative tolerance and 1 × 10−5 absolute tolerance.

that for smaller tµ the strategy works well and good results are obtained. Whentµ = 15, the discontinuity has stopped at x = 0.3 and when a trapezoidal quadrature

rule was used for the source term, a large undershoot and overshoot occurred in the

numerical solution. Leveque and Yee [18] pointed out that the source of difficulty

is the discontinuity in the solution and that a much finer grid is needed there. They

suggested deploying a method that is capable of increasing the spatial resolution near

the discontinuity rather than excessive refinement of the overall grid.

For this purpose a monitor function was used here to guide the decision as to where

to refine or coarsen the mesh. A commonly used monitor function is the second spatial

derivative which, however, tends to infinity around a shock [21] as the mesh is refined.In order to overcome this we have introduced a new monitor function based upon the

local growth in time spatial error est  as defined by Equation (11.13). This leads to

the use of local grid refinement, and with the help of the error balancing approach

described in Section 11.3 it is possible to create a new refined grid directly surrounding

the location of the source. For this purpose we have modified the approach described

by Pennington and Berzins [21]. The remesh routine bisects the mesh cell if the

monitor function is too large or combines two cells into one if the monitor function is

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well below the required value. In the experiments here the remeshing routine is called

on every second time step. The adaptive mesh initially starts with 26 points and when

the error is larger than the specified limit, then the corresponding cell is subdivided

into two with 75 points in total being allowed for the case shown in Figure 11.2, whichshows the front moving correctly. The conclusion from these experiments is that for

problems combining reaction type terms and advection operators, the use of adaptive

mesh techniques within an MOL framework may be a critical factor in ensuring that

a good numerical solution is obtained. The remainder of this chapter will show that

this conclusion also applies to atmospheric modeling problems in two and three space

dimensions.

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1

   S  o   l  u   t   i  o  n

x

Adaptive

True Solution

FIGURE 11.2

True solution (lines) vs. adaptive mesh solution (dots), t  = 0.6.

11.5 Atmospheric Modeling Problem

In order to illustrate the application of the MOL to atmospheric modeling problems,

the model problem considered here involves the interaction of a power plant plume

with background emissions. Such a power plant plume is a highly concentrated sourceof NOx (NO and NO2) emissions, which can be carried through the atmosphere

for hundreds of kilometers, and so provides a stringent test of whether adaptive

gridding methods can lead to more reliable results for complex multi-scale models.

The test conducted here involves considering the interaction of the plume with its

surroundings, and in the model we look at background scenarios of both clean and

polluted air [32]. The test case model covers a region of 300 × 500 km. To keep

the model simple, and therefore reveal particular issues related to the mesh, we have

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used a reduced chemical scheme with idealized dispersion conditions. The domain

is approximated by an unstructured triangular mesh in two space dimensions and

by a tetrahedral mesh in three space dimensions. In both cases the mesh can then

be adapted to higher and higher levels of refinement according to errors in solutioncomponents. The solution technique is based on the spatial discretization of a set of 

advection/diffusion equations on the unstructured mesh using a finite volume, flux-

limited scheme.

The atmospheric diffusion equation in three space dimensions is given by:

∂cs

∂t = −

∂(ucs )

∂x−

∂(wcs )

∂y−

∂(vcs )

∂z+

∂

∂x

Kx

∂cs

∂x

+

∂

∂y

Ky

∂cs

∂y

+ ∂∂z

Kz ∂cs

∂z

+ Rs (c1, c2, . . . , cq ) + Es − (κ1s + κ2s )cs , (11.17)

where cs is the concentration of the sth compound, u, w, and v are wind velocities,

Kx , Ky , and Kz are turbulent diffusivity coefficients, and κ1s and κ2s are dry and

wet deposition velocities, respectively. Es describes the distribution of emission

sources for the sth compound and Rs is the chemical reaction term which may contain

nonlinear terms in cs . For npde chemical species an npde-dimensional set of PDEs

is formed describing the rates of change of species concentration over time and space,

where each may be coupled through the nonlinear chemical reaction terms.In the first instance the restriction to two space dimensions has the advantage that it

is possible to concentrate on showing that standard adaptive numerical methods have

the potential to reveal detail not previously observed in plume models. The extension

to three dimensions will then show that the same conclusions can be drawn but that

there are additional benefits from using mesh refinement vertically.

The simplified chemical mechanism used is shown in Table 1 of Tomlin et al. [32]

and contains only 10 species. Despite its simplicity, it represents the main features of 

a tropospheric mechanism, namely the competition of the fast equilibrating inorganicreactions:

O2

NO2 + hν → O3 + NO

NO + O3 → NO2 + O2,

with the chemistry of volatile organic compounds (voc’s), which occurs on a much

slower time-scale. This separation in time-scales generates stiffness in the result-

ing equations. The voc reactions are represented by reactions of a single species,formaldehyde. This is unrealistic in terms of the actual emissions generated in the

environment, but the investigation of fully speciated voc’s is not the purpose of the

present study. We therefore wished to include the minimum number of reactions

which would lead to the generation of ozone at large distances from the NOx source.

Deposition processes have not been included in the first instance.

In the work of Tomlin et al. [32], the model was used to represent three separate

scenarios of a plume of concentrated NOx emissions being dispersed through a back-

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ground of clean and polluted air. Only one set of these results is shown here. This case

represents a clean air situation where the background levels for NOx and voc’s are

low. Initial conditions for background concentrations are NO2: 1.00 ×108 (molecule

cm−3), NO : 1.00 ×108 (molecule cm−3), O3: 5.00 ×1011 (molecule cm−3),HCHO : 1.00 ×1010 (molecule cm−3). Concentrations in the background change

diurnally as the chemical transformations take place according to photolysis rates,

temperature, and concentration changes.

The power station was taken to be a separate source of NOx and this source was

represented in a slightly different way. In this case, the chimney region is treated as a

subdomain and the concentration in the chimney setas an internal boundary condition.

In terms of the mesh generation, this ensures that the initial grid will contain more

elements close to the concentrated emission source. This is similar in methodology tothe telescopicapproach. Theconcentration in thechimney corresponds to an emission

rate of NOx of 400 kg hr−1. We have considered only 10% of the NOx to be emitted

as NO2.

A constant wind speed of 5 ms−1 in the x-direction was used and the eddy diffusion

parameters Kx and Ky were set at 300 m2s−1 for all species.

11.6 Triangular Finite Volume Space Discretization Method

The basis of the numerical method is the spatial discretization of the PDEs in Equa-

tion (11.17) on unstructured triangular meshes as used in the software SPRINT2D

[7]. The MOL approach then leads to a system of ODEs in time which can then

be solved as an initial value problem, and a variety of powerful software tools exist

for this purpose [5]. For advection-dominated problems it is important to choose a

discretization scheme that preserves the physical range of the solution.

Unstructured triangular meshes are popular with finite volume/element practition-

ers because of their ability to deal with general two-dimensional geometries. In

terms of application to multi-scale atmospheric problems, we are not dealing with

complex physical geometries, but unstructured meshes provide a good method of 

resolving the complex structures formed by the interaction of chemistry and flow in

the atmosphere and by the varying types of emission sources. The term unstructured

represents the fact that each node in the mesh may be surrounded by any number of 

triangles, whereas in a structured mesh this number would be fixed. The discretiza-tion of advection/diffusion/reaction equations on unstructured meshes will now be

discussed.

For systems of equations such as (11.17) it is useful to consider the advective and

diffusive fluxes separately in terms of the discretization. In the present work, a flux-

limited, cell-centered, finite-volume discretization scheme of Berzins and Ware [6, 4]

was chosen. This method enablesaccurate solutions to be determined for both smooth

and discontinuous flows by making use of the local Riemann solver flux techniques

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(originally developed for the Euler equations) for the advective parts of the fluxes,

and centered schemes for the diffusive part. The scheme used for the treatment of the

advective terms is an extension to irregular triangular meshes of the nonlinear scheme

described by Spekreijse [29] for regular Cartesian meshes. The scheme of Berzins andWarehas thedesirableproperties (see Chock [11])of preserving positivity, eliminating

spurious oscillations, and restricting the amount of diffusion by the use of a nonlinear

limiter function. Recent surveys of methods for the advection equation [34, 36] have

suggested the use of a very similar scheme to Spekreijse for regular Cartesian meshes,

preferring it to schemes such as flux-corrected transport.

To illustrate this method, consider the advection-reaction equation that extends

Equation (11.1) to two space dimensions:

∂c

∂t = −

∂uc

∂x−

∂wc

∂y+ R(c) , t  ∈ (0, t e),(x,y) ∈ (11.18)

with appropriate boundary and initial conditions. A finite volume type approach is

adopted in which the solution value at the centroid of triangle i, (xi , yi ), is ci  and

the solutions at the centroids of the triangles surrounding triangle i are cl , cj  , and

ck . Integration of Equation (11.18) on the ith triangle, which has area Ai , use of the

divergence theorem, and the evaluation of the line integral along each edge by the

midpoint quadrature rule gives an ODE in time:

dci

dt = −

1

Ai

ucik y0,1 − vcik x0,1 + ucij y1,2

− vcij x1,2 + ucil y2,0 − vcil x2,0

+ R(ci ) , (11.19)

where xij  = xj  − xi , yij  = yj  − yi . The fluxes ucij   and vcij   in the x and y

directions, respectively, are evaluated at the midpoint of the triangle edge separating

the triangles associated with ci and cj  . These fluxes are evaluated by taking account

of the flow directions with respect to the orientation of the triangle. This is achievedby using either the left or right solution values depending on the direction of advection

and how each edge is aligned. These left and right solution values for each edge in

a triangle are defined as the left solution value being that internal to the ith triangle,

and the right  solution value being that external to triangle i. Consider, for example,

the case shown in Figure 11.3 when u is positive and xi < xj . This means that the

x component of the advection is flowing from node i to node j , and so cij  = clij .

Similarly when v is positive the y component of the wind is blowing from node k to

node i and so cik = crik . Hence, Equation (11.19) may be written as

dci

dt = −

1

Ai

ucl

ik y0,1 − vcrik x0,1 + ucl

ij y1,2

− vclij x1,2 + ucr

il y2,0 − vclil x2,0

+ R(ci ) . (11.20)

A simple first-order scheme uses clij  = ci , cr

ij  = cj  on the edge between triangles

i and j . This scheme is too diffusive and so Berzins and Ware [6] use a complex

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interpolation scheme to obtain the left  and right  values on each edge. The inter-

polants in this second order scheme use a constrained or limited form of the solution

obtained from the six triangles surrounding an edge giving a 10-triangle stencil for

the discretization of the convective terms on each triangle. For example, the value

interpolated solution values

centroid solution values

midpoints of edges

q

pqc

c

c il

 j

ij

c

pc

nc

mc

rsc

r

c

cmnc l

ck 

c

c

s

ik 

c

ljc

kjc

lk c

ic

0(X ,Y )0

2(X ,Y )

1

2

(X ,Y )1

FIGURE 11.3

Interpolants used in irregular mesh flux calculation.

clij  is constructed by forming a linear interpolant using the solution values ci , ck , and

cl at the three centroids. An alternative interpretation is that linear extrapolation isbeing used based on the solution value ci and an intermediate solution value (itself 

calculated by linear interpolation) clk which lies on the line joining the centroids at

which cl and ck are defined (see Figure 11.3), i.e.,

clij  = ci + (S ij ) d ij,i

ci − clk

d i,lk, (11.21)

where the argument S  is a ratio of solution gradients defined in a way similar to the

ratio rj  in Equation (11.5), see [6], and the generic term d a,b denotes the positivedistance between points a and b. For example d ij,i denotes the positive distance

between points ij  and i, see Figure 11.3, as defined by

d i,ij  =

 (xi − xij )2 + (yi − yij )2 , (11.22)

where (xij , yij ) are the coordinates of  cij . In order to preserve positivity in the

numerical solution, the limiter function is used and has to satisfy (S)/S  ≤ 1,

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see [6]. These conditions are satisfied, for example, by a modified van Leer limiter

defined by:

(S) = (S + |S |)/(1 + Max(1, |S |)) . (11.23)

The value crij  is defined in a way similar to using the centroid values cj , cs , and cr .

This scheme is of second-order accuracy, see [6]. The diffusion terms are discretized

using a finite-volume approach to reduce the integrals of second derivatives to the

evaluation of first derivatives at the midpoints of edges. These first derivatives are

then evaluated by differentiating a bilinear interpolant based on four midpoint values,

see [7]. Theboundary conditions are implemented by includingthem in thedefinitions

of the advective and diffusive fluxes at the boundary.

11.7 Time Integration

An MOL approach with the above spatial discretization scheme results in a system

of ODEs in time which are integrated using the code SPRINT [5] with the Theta

or BDF options which are specially designed for the solution of stiff systems withmoderate accuracy and automatic control of the local error in time. Once the PDEs

have been discretized in space we are left with a large system of coupled ODEs of 

dimension N  = m × npde where m is the number of triangles in the mesh, and npde

the number of species. These equations may now be written in the same form as

Equation (11.2) as

c = F N ( t, c(t) ), c(0) given , (11.24)

where, in the caseofa singlespecies, the vector, c(t), isdefined byc(t) = [c(x1, y1,t),. . . , c ( xN , yN , t)]T . The point xi , yi is the center of the ithcell and Ci (t) is defined as

a numerical approximation to the exact solution to the PDE evaluated at the centroid,

i.e., c(xi , yi , t). The MOL approach is used to numerically integrate Equation (11.24)

thus generating an approximation, C(t), to the vector of exact PDE solution values

at the mesh points, c(t).

The Theta method [6], which has been used for the experiments described here,

defines the numerical solution at t n+1 = t n + t , where t  is the time-step size, as

denoted by C(t n+1), by:

C(t n+1) = C(t n) + (1 − θ )t  C(t n) + θ  t F N (t n+1,C(t n+1)) , (11.25)

in which C(t n) and C(t n) are the numerical solution and its time derivative at the

previous time t n and θ  = 0.55. This system of equations is solved by using the

approach described in Section 11.2. In this case, the matrix J s is block-diagonal with

as many blocks as there are triangles and with each block having as many rows and

columns as there are PDEs. The fact that the blocks relate only to the chemistry

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plus source/sink terms within each cell, means that the equations may be solved

independently using LU decomposition, or even more efficiently by using Gauss–

Seidel iterations, see [35]. This approach mayalsobe interpreted as approximating the

flow term [I −tθ J f ] by theidentitymatrix, as isdone when using functional iterationwith the Theta method applied to flow alone [3]. Since the spatial discretization

method connects each triangle to as many as 10 others, it follows that the matrixI  − tθ J f 

may have a much more complex sparsity pattern than that of the block-

diagonal matrix [I  − tθ J s ]. Approximating the matrix

I  − tθ J f 

by the identity

matrix [6] thus eliminates a large number of the full Jacobian entries. Moreover, the

use of Gauss–Seidel iteration makes it possible to solve these problems without any

matrices being stored. This approach is particularly useful in three space dimensional

problems.The original approach of Berzins [3] was only extended to source-term problems

by Ahmad and Berzins [1]. As a consequence the calculations performed by Tomlin

et al. [32] used the standard local error approach given by:

|| le(t n+1) || < T OL , (11.26)

where le is the local error defined as in Equations (11.12) and (11.13).

11.8 Mesh Generation and Adaptivity

The initial unstructured meshes used in SPRINT2D are created from a geometry

description using the Geompack [16] mesh generator. These meshes are then refined

and coarsened by the Triad adaptivity module, which uses data structures to enable

efficient mesh adaptation.Since the initial mesh is unstructured we have to be very careful in choosing a

data structure that provides the necessary information for refining and derefining

the mesh. When using a structured mesh it is possible to number mesh vertices

or elements explicitly. This is not possible for unstructured meshes and therefore

the data structure must provide the necessary connectivity. The important factor is

to maintain the quality of the triangle as the mesh is refined and coarsened. This

is achieved using a tree-like data structure with a method of refinement based on

the regular subdivision of triangles. These may later be coalesced into the parent

triangle when coarsening the mesh. This process is called local h-refinement, sincethe nodes of the original mesh do not move and we are simply subdividing the original

elements. Three examples of adaptive meshes for a single moving front at different

times are shown in Figure 11.4. These meshes show how the adaptive mesh follows

the front as it moves in time across the spatial domain. Similar procedures are used

extensively with a wide range of both finite element and volume methods for a very

broad range of physical problems. Once a method of refinement and derefinement

has been implemented, it remains to decide on a suitable criterion for the application

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FIGURE 11.4

Sequence of refined meshes.

of the adaptivity. The ideal situation would be that the decision to refine or derefine

would be made on a fully automatic basis with no user input necessary. In practice a

combination of an automatic technique andsome knowledgeof thephysical properties

of the system is used. The technique used in this work is based on the calculation of 

spatial error estimates. Low- and high-order solutions are obtained and the difference

between them gives the spatial error, as in Section 11.3 and in [3] but without the

extension to source terms in [1]. The algorithm can then choose to refine in regions

of high spatial error by comparison with a user defined tolerance. For the ith PDE

component on the j th triangle, a local error estimate ei,j (t) is calculated from the

difference between the solution using a first-order method and that using a second-order method. For time-dependent PDEs this estimate shows how the spatial error

grows locally over a time step. A refinement indicator for the j th triangle is defined

by an average scaled error (serrj ) measurement over all npde PDEs using supplied

absolute and relative tolerances:

serrj  =

npdei=1

ei,j (t)

atoli /Aj  + rtoli × Ci,j 

, (11.27)

where atol and rtol are the absolute and relative error tolerances. This formulation

for the scaled error provides a flexible way to weight the refinement towards any PDE

error. An integer refinement level indicator is calculated from this scaled error to

give the number of times the triangle should be refined or derefined. Since the error

estimate is applied at the end of a time step, it is too late to make the refinement

decision. Methods are therefore used for the prediction of the growth of the spatial

error using linear or quadratic interpolants. The decision about whether to refine a

triangle is based on these predictions, and the estimate made at the end of the time

step can be used to predict errors at future time steps. Generally it is found thatlarge spatial errors coincide with regions of steep spatial gradients. The spatial error

estimate can also be used to indicate when the solution is being solved too accurately

and can indicate which regions can be coarsened.

For applications such as atmospheric modeling it is important that a maximum

level of refinement can be set to prevent the code from adapting to too high a level

in regions with concentrated emissions. This is especially important around point or

highly concentrated area sources. Here, because of the nature of the source, steep

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spatial gradients are likely to persist down to very high levels of refinement. This

would have the consequence that the number of elements on which the PDEs had to

be discretized would become prohibitively large. For the following test problems the

maximum level of refinement was therefore limited to level 3.

11.9 Single-Source Pollution Plume Example

The example used here to illustrate the effectiveness of the adaptive mesh is that

of a single plume pollution source. In this case the initial two-dimensional mesh wasgenerated with only 100 elements. It is difficult to relate the size of unstructured

meshes directly to regular rectangular ones, but our original mesh was comparable

to the size of mesh generally used in regional scale atmospheric models, the largest

grid cell being approximately 60 km along its longest edge. Close to the chimney the

mesh was refined to elements of length 5 km ensuring that it would be refined to a

reasonable resolution in this region of steep gradients. If we allow the mesh to refine

two levels, then the smallest possible mesh size close to the chimney will be 1.25 km

in length. Spatial errors in the concentration of NO were chosen as the criterion from

which to further refine the mesh. Test runs showed that regions of high spatial errorcoincided with steep spatial gradients. The mesh can therefore be considered to adapt

around steep NO concentration gradients. Each run was carried out over a period of 

48 h starting from midnight on Day 1, so that the diurnal variations could be observed.

We present here only a selection of the results that illustrate the main features relating

to the mesh adaptation.

Figures 11.5 and 11.6 allow a comparison to be made between the structure of 

FIGURE 11.5

The structure of the level 0 mesh. The length of the domain is 300 km and the

width 200 km. The smallest and largest mesh lengths are approximately 5 and

60 km, respectively, for the level zero domain.

the base mesh and a mesh that has been adapted up to level 2 at 14.00 on Day 2.

In these figures the sides of the polygons represent the distance between cell centers

on the triangular mesh. The main area of mesh refinement is along the plume edges

close to the chimney, indicating that there is a high level of structure in these regions.

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FIGURE 11.6

The structure of the level 2 adaptive mesh.

On the coarse mesh the plume is dispersed over a much larger area than on the finemesh and most of the plume structure is lost. Close to the stack the concentration of 

O3 is much lower than that in the background because of high NOx concentrations.

The inorganic chemistry is dominant in this region and the ozone is consumed by the

reaction: NO + O3 → NO2 + O2.

In Figure 11.7 we present a cross-plume profile of the NO2 concentrations at a

0

1e+11

2e+11

3e+11

4e+11

5e+11

6e+11

7e+11

8e+11

9e+11

1e+07 1.2e+07 1.4e+07 1.6e+07 1.8e+07 2e+07

   N   O   2  c  o  n  c  e  n   t  r  a   t   i  o  n

Displacement in y

level-0

level-3

FIGURE 11.7

Cross plume NO2 profiles 10 km from stack in molecules cm−3, showing how

the level 3 solution captures the structure of the plume.

distance of 10 km downwind of the chimney stack for Case A at the same time as

the previous figure. The figure clearly shows the features at the edge of the plume

which are revealed by the adaptive solution. From the base mesh, where the distancebetween elements along the y-axis close to the stack is 20 km, it appears that the

concentration of NO2 rises to a peak in the center of the plume. If the mesh is refined

to higher levels, then we start to see the true structure of the plume emerging. With

a level 3 solution we can see that the peak concentrations are actually found along

the edges of the plume and that the concentration of NO2 drops to very low levels at

the plume center. From the area under these curves it is found that there is a 30%

difference between the overall level 0 and the level 3 concentrations. This shows

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that not only the peak concentrations, but the total integrated concentrations are very

different for the different levels of mesh adaptation. It is clear therefore that using a

very coarse grid in regions of steep spatial gradients can lead to an over-estimate of 

total pollutant concentrations for systems with nonlinear chemical schemes.Figures 11.8 and 11.9 show that in the case considered here the plume is over-

0.0x100 1.0x107 2.0x107 3.0x107 4.0x107 5.0x107

0.0x100

1.0x107

2.0x107

3.0x107

Level 0Clean air

    7.   3

   9.  4 1 1.

 5

1     3     . 6      

1 5. 7

 1 5. 717 .8 

 1 7. 8

1 7. 8

1  7  .8   17 .8

1 9. 9

19 .9 

 1 9. 9

19.91 9 .9 

17 .8

FIGURE 11.8

Ozone contours for Case C, clean air, level 0 calculation.

0.0x100 1.0x107 2.0x107 3.0x107 4.0x107 5.0x107

0.0x100

1.0x107

2.0x107

3.0x107

Level 2Clean air

1

3.15 .2

5.2

7 .3

 7. 3

9 .4

9   . 4     9. 4

1 1 .5   11.5

11.5

13.6

1   3    . 6    

13.6

15.71 5  .7  

1  5.  7

15. 715.7

17 .8 

 1 7. 8

 1 7. 8

17 .8 

17.8

 1 9. 9

1 9. 9

1 9  .9  

19.9

1 9. 9

1 9. 9

 7. 3

FIGURE 11.9

Ozone contours for Case C, clean air, level 2 calculations.

dispersed in the level 0 case and the spatial distribution of ozone is therefore inaccu-rately represented. For the clean air case, the levels of ozone drop considerably in the

plume compared to the background since the levels of NO are much higher there. For

the level 0 case these lowered concentrations spread over very large distances owing

to the over-dispersion of the plume. The location of reduced/raised concentrations

will therefore be incorrect for the level 0 results in all three cases. For each scenario,

the level 0 solution leads to a smoothing out of the ozone profiles so that the true

structure caused by the interaction of the plume with background air is missed.

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The striking result is that the adaptive solution reveals features such as peak levels

of NO2 and O3 which could not be detected using a coarse mesh. The change in mesh

refinement also resulted in a change in overall or integrated concentration levels.

This indicates that due to strongly nonlinear terms in the chemical reaction rates,the source terms in the PDE will be mesh dependent. Without using a fine mesh

over the whole domain so that the concentrations in neighboring cells differ only

very little, the effects of this nonlinearity could be quite significant. To reduce the

effects it is important to refine the mesh at least in regions of steep spatial gradients.

This has been partially addressed by the telescopic methods presently used in air

quality models. However, the present test case has shown that steep gradients can

occur at long distances downwind from the source, for example the change in ozone

concentrations along the edges of the plume. Adaptive algorithms seem to present asuccessful method of achieving accuracy in such regions and can do so in an automatic

way. The main limitation of the above approach is that only two space dimensions

have been considered. The next issue to be resolved is whether mesh adaptation is

necessary in the vertical direction and how appropriate an MOL approach is in three

space dimensions. These are the issues considered in the next five sections.

11.10 Three Space Dimensional Computations

The standard approach with three space dimensional atmospheric dispersion prob-

lems is that in the vertical domain usually a stretched mesh is used, placing more

solution points close to the ground. As in the horizontal domain, the resolution of 

the mesh in the vertical direction affects the vertical mixing of pollutant species. The

use of adaptive meshes in the vertical domain has thus far received little attention.

In the work described here we have used two approaches for solving three space

dimensional atmospheric dispersion problems. Both approaches use a fully 3D un-

structured mesh based on tetrahedral elements. The first approach is described in [17]

and is the closer of the two approaches to the two-dimensional case described in

Section 11.6 in that a cell-centered, finite-volume scheme is used for the spatial dis-

cretization. In this case a conventional MOL approach is used based on a modified

version of the SPRINT time-integration package. The linear algebra approach of 

Section 11.7 is used with a simple first-order spatial discretization approach. Thedisadvantage of this approach is that it requires a much larger number of unknowns

for a given mesh than if a cell-vertex approach is used with the solution unknowns

being positioned at the nodes of the mesh. The price that is paid for this reduction in

the number of unknowns is an increase in the complexity of the discretization method.

There is also the well-known difficulty that the cell-vertex discretization may not pre-

serve the positivity of the solution on certain meshes due to the discretization of the

diffusion operator [9]. Although it may be possible to address this issue within an

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MOL framework, the need to preserve positivity and the different time-scales needed

for advection and chemistry have led us to employ an operator-splitting approach.

The next section describes the 3D unstructured mesh discretization method and the

flow-integration scheme which advances the solution in time. Section 11.12 containsthe mesh-adaptation strategy which changes the connectivity in the data structure

of the mesh in response to changes in the solutions. Section 11.13 explains the

implicit-explicit method used to solve the transport equation. Section 11.14 contains

the test examples which have been designed to determine the importance of mesh

structure on both horizontal and vertical mixing for typical meteorological conditions.

The test problem describes the dispersion of pollutants from a single source due to

typical boundary-layer wind profiles. Finally, we draw conclusions in Section 11.15

regarding the importance of adaptive-mesh method in solving 3D atmospheric-flowproblems.

11.11 Three Space Dimensional Discretization

The atmospheric-diffusion equation is discretized over special volumes that form

the dual mesh. The dual mesh is formed by constructing non-overlapping volumes,referred to as dual cells, around each node. The dual mesh for a tetrahedral grid is

constructed by dividing each tetrahedron into four hexehedra of equal volumes, by

connecting the mid-edge points, face-centroids, and the centroid of the tetrahedron.

The control volume around a node 0 is thus formed by a polyhedral hull which is

the union of all such hexahedra that share that node. The quadrilateral faces that

constitute the dual mesh may not all be planer. Each component of the diffusion

Equations (11.17) is discretized using thesame method. Hence, for simplicity, instead

of treating the vector c, we choose one of its components, say c, and describe itsdiscretization.

11.11.1 Flux Evaluation Using Edge-Based Operation

The evaluation of flux around a dual cell can be cast in an edge-based operation.

Let us discretize the divergence term

∂f 

∂x+

∂g

∂y+

∂h

∂z

over the control volume 0 enclosing the node 0. This divergence form is converted

to flux form using Gauss divergence theorem: 0

∂f 

∂x+

∂g

∂y+

∂h

∂z

d =

 ∂0

(f nx + g ny + h nz) dS 

=

k

f S x + g S y + h S z

, (11.28)

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where the summation is over all the dual mesh faces that form the boundary of the

control volume around the node 0 and the areas S x , S y , S z are projections of the dual

quadrilateral face.

Consider edge i, formed by nodes 0 and N(i). The quadrilateral faces of the dualmesh that are connected to the edge at its mid-point P are shown in Figure 11.10.

The number of such quadrilateral faces attached to an edge depends on the number

1

2 3

4

0

N(i)

P

FIGURE 11.10

Dual mesh faces attached to an edge.

of tetrahedra neighbors to that edge. There are four tetrahedra sharing the edge i in

Figure 11.10. The projected area, Ai , associated with the edge i is calculated in terms

of the quadrilateral face areas, a1, a2, a3, a4, as

(Ai )x =

4j =1

(aj )x , (Ai )y =

4j =1

(aj )y , (Ai )z =

4j =1

(aj )z . (11.29)

The projections are computed so that the area vector points outward from the controlvolume surface associated with a node. The boundary of the control volume around

the node 0 is formed by the union of all such areas Ai associated with each edge i

that shares the node 0. The contribution of the edge i to the fluxes across the faces of 

the control volume surrounding the node 0 is given by

f p (Ai )x + gp (Ai )y + hp (Ai )z .

Hence, Equation (11.28) is replaced by

 0

∂f 

∂x+

∂g

∂y+

∂h

∂z

d =

i

f p (Ai )x + gp (Ai )y + hp (Ai )z

, (11.30)

where the sum is over the edges that share the node 0. The fluxes are thus calculated

on an edge-wise basis and conservation is enforced by producing a positive flux

contribution to one node and an equally opposite contribution to the other node that

forms the edge.

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11.11.2 Adjustments of Wind Field

In an atmospheric pollution model, we often use observed wind data which are not

mass conservative. Even mass-conserving wind data might not be mass conservativein the numerical sense when interpolated onto an unstructured grid. Thus, we want

to adjust the wind data in such a way that the observed data are minimally changed

while still satisfying the mass-conservative property numerically. If u , v , w are the

wind velocities, then they must satisfy

∂u

∂x+

∂v

∂y+

∂w

∂z= 0 . (11.31)

Here we enforce mass conservation using the variational calculus technique of Mathur

and Peters [20]. The technique attempts to adjust the wind velocity in a manner suchthat the interpolated data are minimally changed in a least-squares sense, and at the

same time, the adjusted values satisfy the mass-conservation constraint. The details

are provided by Ghorai et al. [12].

We have adjusted one-dimensional stable, neutral, and unstable boundary layer

wind velocities which are a function of z. The wind velocity is mass conservative

analytically. The wind velocity remains mass conservative in the numerical sense

in the base mesh since the unstructured base mesh is regular, but may not be nu-

merically mass conservative once the grid is refined (derefined). A representativeone-dimensional neutral boundary layer velocity is shown in Figure 11.11(b). The

velocity field is adjusted in the refinement region, but away from the refinement

region, the velocity remains almost unchanged.

FIGURE 11.11

A representative variation of wind with height for (a) stable, (b) neutral, and (c)

unstable boundary layers.

The base grid spacings along the vertical increases upwards. Thus, the grid quality

near the ground is worse due to the large aspect ratio of the tetrahedron. The velocity

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corrections decrease upwards as the refine region moves upwards. Suppose we have a

refine region at 150 m height. The maximum corrections are 12, 14, and 0.06 cm s−1,

respectively, for the u, v, and w components. For a refine region at 600 m height the

corresponding components are 11, 11, and 0.03 cm s−1. And finally, the correspond-ing corrections decreases to 0.3, 0.2,and0.0002 cm s−1 at 1.8 km height. The neutral

boundary layer velocity increases from 0 to 9 m s−1 as z increases from 0 to 3 km

and so the velocity corrections are small.

11.11.3 Advection Scheme

The discretization of the term

 0

∂(uc)

∂x+

∂(v c)

∂y+

∂(w c)

∂z

d ≡

 0

∇ .F d ,

where

F  =

ui + vj  + wk

c ,

is done by using an algorithm based on that of Barth and Jespersen [8] and uses

Equation (11.30) to rewrite the equation above as an edge-based computation:

i

up (Ai )x + vp (Ai )y + wp (Ai )z

(c)p =

i

F p.Ai

(c)p , (11.32)

where Ai is called the edge-normal associated with the edge i and the sum is over all

the edges sharing the node 0 with control volume 0. Evaluation of this expression isbyusing the upwind limited approach of Barth and Jesperson[8]. The values of limiter

functions and gradient at the nodes are not calculated on a node-by-node basis (which

is CPU intensive), instead they are calculated in an edge-based operation [8, 9]. The

time step for the advection scheme is chosen so that it satisfies the CFL condition [37].

The minimum of the time steps over all the vertices constitutes the time step for the

advection scheme. Again this computation can be cast into an edge-based operation.

11.11.4 Diffusion Scheme

The diffusion term is discretized using the standard linear finite-element method

or the equivalent cell-vertex method described by Barth. Again the key feature is that

the calculation of the diffusion terms is reordered so that it involves edge gradient

terms, see [9]. The disadvantage of the standard approach is that it does not preserve

positivity of the solution for certain meshes, see [9]. Very recent work has provided

methods that begin to address this issue [23].

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11.12 Mesh Adaptation

The cell-vertex scheme approach is hierarchical in nature [10, 28], and is applica-

ble to meshes constructed from tetrahedral shaped elements. The basic mesh objects

of nodes, edges, faces, and elements, that together form the computational domain,

map onto the data objects within the adaptation algorithm tree data structure. The

data objects contain all flow and connectivity information sufficient to adapt the mesh

structure and flow solution by either local refinement or derefinement procedures. The

mesh-adaptation strategy assumes that there exists a “good quality” initial unstruc-

tured mesh covering the computational domain. The refinement process adds nodesto this base level mesh by edge, face, and element subdivision, with each change

to the mesh being tracked within the code data structure by the construction of a

data hierarchy. The derefinement is the inverse of refinement, where nodes, faces,

and elements are removed from the mesh by working back through the local mesh

refinement hierarchy.

The main adaptation is driven by refining and derefining element edges. Thus, if 

an edge is refined by the addition of a node along its length, then all the elements that

share the (parent) edge under refinement must be refined. In the case of derefinementall the elements that share the node being removed must be derefined. Numerical

criteria derived from the flow field will mark an edge to either refine, derefine, or

remain unchanged. It is necessary to make sure the edges targeted for refinement

and derefinement pass various conditions prior to their adaptation. These conditions

effectively decouple the regions of mesh refinement from those of derefinement,

meaning that, for example, an element is not both derefined and refined in the same

adaptation step.

For reasons of both tetrahedral quality control and algorithm simplicity only two

types of element subdivisions are used [28]. The first type of subdivision is called

regular subdivision, where a new node bisects each edge of the parent element re-

sulting in eight new elements. The second type of dissection, green subdivision,

introduces an extra node into parent tetrahedron, which is subsequently connected

to all the parent vertices and any additional nodes that bisect the parent edges. The

green refinement inconsistently removes connected or “hanging” nodes without the

introduction of additional edge refinement. The green elements may be of poorer

quality in terms of aspect ratio and so the green element may not be further refined.

Figure 11.12 demonstrates regular and green refinement for a tetrahedron. The fivepossible refinement possibilities (if all the edges are refined then the parent element

is regularly refined) give rise to between 6 and 14 child green elements.

The choice of adaptation criteria is very important since it can produce either a

large or small number of nodes depending on the condition used to flag an edge for

the adaptation. Also, when there are a large number of species, the choice of a given

criteria might result in high resolution for some species but low resolution for the

other species. Let 0 and i be the nodes for a given edge e(0, i). We calculate tolg

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FIGURE 11.12

(a) Regular refinement based on the subdivision of tetrahedron by dissection

of interior diagonal (1:8) and (b) “green” refinement by addition of an interior

node (1:6).

and tolc by

tolg =|(c)0 − (c)i |

distand tolc =

(c)0 + (c)i

2,

where dist is the length of the edge e(0, i). We refine the edge e(0, i) if tolg and

tolc exceed some tolerances, otherwise it is derefined. Also a maximum level of 

refinement is specified at the beginning so that if an edge is targeted for refinement

but it is in the maximum level, then it is kept unchanged.

Suppose we have two edges with tolg = 100 and 200. If we take the toleranceparameter, T g say, for tolg equal to 150, then only the second edge is refined to

maximum level. On the other hand, if  T g = 50, then both edges are refined to

maximum level. We expect that the solution error for edge with tolg = 200 is greater

than the error in the edge with tolg = 100. It might be advantageous to use two sets

of T g = 50 and 150. If tolg > 150, than we refine an edge to maximum level and

if 50 < tolg < 150, then we refine an edge to the level just lower than the maximum

levels. Thus, the idea is to refine to the maximum level in the steepest gradient regions

but to lower levels in the regions of less steep gradients.

11.13 Time Integration for 3D Problems

Although in two space dimensional calculations we have used sophisticated space-

time error control techniques [7, 32], the need to preserve positivity, to reduce com-

putational cost, and to take into account the different time-scales needed for the inte-gration of advection and chemistry has led us to use an operator-splitting technique.

In this approach, the chemistry is decoupled from the transport. The main reason for

the use of this is that it is much easier to ensure positivity of the solution components.

The nonlinear chemistry part gives rise to stiff ordinary differential equations. We

solve the chemistry part using the SPRINT time integration methods [7] and have also

used the Gauss–Seidel iteration of Verwer [35]. The transport step is considered first.

If cn denotes the species concentration at time level t n, then the species concentration

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at the next time step is given by

cn+1 = cn + tg(c) + tf(c) + S , (11.33)

where t  is the time step, g(c) is the advection operator, and f(c) is the diffusion

operator. In a fully explicit scheme, f  and g are evaluated using values at the time

level n. However, the time restriction for stability due to vertical diffusion is severe

since the grid spacings along the vertical can be small. Hence, we use an implicit-

explicit formulation for Equation (11.33), where the advection is evaluated explicitly

and the diffusion is calculated implicitly. Again let us consider node i and let N(i) be

the set of nodes sharing the node i. The discretized form of the advection-diffusion

equation for c at node i is given by

1

t + ai

(cn+1)i =

j ∈N(i), j =i

aj 

cn+1

j 

+ Qni , (11.34)

where i is varied over all the nodes and

Qi =

cn

t + g(cn) + S 

i

.

The time step t  is chosen to be equal to the time step due to advection only. The

value of time step mainly depends on the wind speed and the vertical mesh spacings

near the source. For the base mesh (described in the next section) used in the test

examples, t  is ≈ 35 s for the stable atmospheric boundary layer but decreases to

≈ 18 s for the unstable atmospheric boundary layer. Thus, the time step is smaller for

higher wind speed and vice versa. The system of equations given by Equation (11.34)

is solved using the Gauss–Seidel iteration technique with over-relaxation and the

iteration is stopped when the relative error is less than some prescribed tolerance.

The advantage of this method is its computational efficiency. The disadvantage is

that we are introducing an extra time integration and splitting error which is not

easily quantified. In future work we will revisit this issue of a standard method of 

lines approach vs. the operator-splitting approach used here.

11.14 Three-Dimensional Test Examples

The advection schemehas been tested byadvecting a puff ofNOaround a horizontal

circle without any diffusion [33]. The results showed that the peak almost remains

constant suggesting that very little artificial diffusion has taken place for refined

meshes. Here we consider the solution of the combined advection-diffusion problem

with a source term that relates to the long-range transport of a passive species from

an elevated point source.

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FIGURE 11.13

A representative mesh for the 3D atmospheric dispersion problem.

The background concentration of NO is 7.5 × 1010 molecules/cm3. The horizontal

dimensions of the domain are 96 km and 48 km along the x and y axis, respec-

tively. The vertical height of the domain is 3 km. We consider a point source at

(6, 24, 0.24) km location with an NO emission rate of 1.98 × 1024

molecules s−1

.For simplicity, we consider a constant wind direction along the x-axis. We consider

three different wind velocity and vertical diffusion profiles which are representative

of stable, neutral, and unstable boundary layers. The corresponding velocities and

vertical diffusions are shown in Figures 11.11 and 11.14 from Seinfeld [25].

FIGURE 11.14

A representative variation of vertical diffusion with height for (a) stable, (b)

neutral, and (c) unstable boundary layers.

The horizontal diffusion coefficients Kx and Ky are kept constant and equal to

50 m2, s−1. The initial tetrahedral mesh is generated by dividing the whole region

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into cuboids and then subdividing a cuboid into 6 tetrahedral elements. The cuboids

are 4 km and 4 km along the x and y axis, respectively. The vertical height is divided

into nine layers and the layers are placed at 0, 0.206, 0.460, 0.767, 1.13, 1.54, 2.0,

2.45, and 3 km heights, respectively.We compute the solutions on the adaptive grid and also check the accuracy against

a reference solution. The reference solution is obtained on a fixed grid generated

from the base mesh by refining all the edges (to level 3) which lie inside a box lying

along the x-axis through the source. We also compute the solution on a telescopic grid

with refinement around the source and compare the solution with the adaptive and

reference solution. The vertical turbulent diffusivity coefficient is small and confined

very near to the ground level for the stable boundary layer. Thus, the concentration

does not mix much above the source height. The height of the reference box is1/2 km and the width is 10 km for the stable boundary layer. On the other hand, the

pollutant becomes well mixed above the source height for the neutral and unstable

boundary layers. Thus, a box of width 10 km and height 1 km is chosen for the

neutral and unstable boundary layers. The total number of nodes in the reference grid

is 114,705 for the stable layer and 142,247 for the neutral and unstable boundary

layers. The initial grid for the adaptive solution is generated by refining a region

around the point source. The refinement region lies horizontally within a 3-km circle

with the point source as the center and it lies vertically within 300 m from the source.

The initial number of nodes is 6,442 for all three boundary layers. The number of nodes for the telescopic method remains 6,442 throughout the simulation period. On

the other hand, the adaptive grid is refined/derefined as the solution advances. The

time step t  for the implicit-explicit scheme is small (usually less than 1 min) due to

small vertical spacings near the ground level which affect the CFL condition. Instead

of carrying out the adaptation after every time step (which is CPU intensive), the

adaptation is carried out approximately every 20 min. This prevents large amounts of 

computational effort being used to perhaps refine very few tetrahedra each time step

and does not significantly affect solution accuracy.

11.14.1 Grid Adaptation

Three sets of tolerance parameters are chosen for the adaptive grid method for each

boundary-layer profile as described below. Let TOLg be the maximum values of 

tolg outside the source region. The refinement criteria of the edges are

(a) Refine edges to level 3 if tolc > 9 × 1010 and tolg > 0.002 × TOLg

(b) Refine edges to level 2 if tolc > 9 × 1010 and tolg > 0.00002 × TOLg

(c) Refine edges to level 1 if tolc > 9 × 1010 and tolg > 0.000001 × TOLg

for the stable boundary layer.

The corresponding criteria for the neutral and unstable boundary layers are

(a) Refine edges to level 3 if tolc > 1011 and tolg > 0.01 × TOLg

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(b) Refine edges to level 2 if tolc > 1011 and tolg > 0.0005 × TOLg

(c) Refine edges to level 1 if tolc > 1011 and tolg > 0.00005 × TOLg

The total number of nodes generated by the adaptive grid method are 60,000,

51,000, and 52,000 for the stable, neutral, and unstable boundary layers, respectively.

The adaptive grid refinement in the vertical plane downwind along the plume center-

line is shown in Figure 11.15. The concentration is confined near the ground level due

(a)

(b)

(c)

 y=24 (km)

 x  (km)

      z    (   k

  m   )

FIGURE 11.15

Grid refinement in the vertical plane through the source along the downwind

direction for the (a) stable, (b) neutral, and (c) unstable boundary layers.

to small vertical diffusion for the stable case. This produces high spatial gradientswithin this region and grid refinement is highest near the ground. Since the vertical

diffusion for the other two cases is larger compared to the stable boundary layer, the

grid refinement extends to almost 1 km from ground level. It is also interesting to

note that at large distances downwind from the source, the adaptive technique places

more mesh points at the top of the boundary layer domain. This reflects the steep

gradients found here due to a significant drop in the vertical diffusion coefficient Kz.

This result may have significance for models attempting to represent boundary-layer

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transport and mixing since the usual approach to vertical meshing is to place a greater

number of mesh points close to the ground and not the top of the boundary layer.

For the unstable boundary layer [see Figure 11.15(c)], the concentration becomes

uniformly mixed below the inversion layer but very little diffusion is taking placeabove the inversion layer. The gradient is high near the inversion layer compared

to the gradient near the ground. Thus, the edges near the inversion layer refine to a

higher level than the edges near the ground.

The adaptive grid refinement at three different locations in the cross-wind direction

is shown by Ghorai et al. [12]. The concentration gradients remain high for the stable

case but low for the neutral and unstable cases far downwind from the source. Thus,

the edges for the stable boundary layer, far downwind from the source, are refined to

higher level than for the neutral and unstable cases. The gradients are high near thesource for all the three cases and the edges are refined to the maximum level for all

of them.

11.14.2 Downwind Concentration

The solutions downwind along the plume center-line in the ground level are shown

in Figure 11.16. The maximum relative errors with respect to reference solutions are

16%, 20%, and 20% approximately for the stable, neutral, and unstable boundarylayers, respectively. The maximum errors for the neutral and unstable cases occur

far downwind from the source where the magnitude of the concentrations are small.

The solution on the telescopic grid is accurate near the source region only due to

the refinement in this region. Far downwind from the source, the solution on the

telescopic grid differs widely from the reference solution. The programs have been

run serially on an Origin2000 computer. For the neutral boundary layer, the total CPU

times are approximately 1, 7, and 25 h for the telescopic, adaptive, and reference grids,

respectively. Thus, the adaptive method is efficient compared to the other methods

and achieves greater accuracy in a reasonable time.

11.15 Discussions and Conclusions

In this chapter we have described an MOL approach to the solution of transient

reacting-flow problems. In particular, the atmospheric diffusion equation was solvedby using unstructured, adaptivemeshes with the MOL in two space dimensions. How-

ever, because of efficiency and positivity considerations, the three space dimensional

case was solved by using operator splitting. The single most important conclusion is

that there are key features of plume characteristics which cannot be represented by

the coarse meshes generally used in regional scale models.

The test cases have demonstrated that adaptive methods can give much improved

accuracy when compared to telescopic refinement methods particularly at large dis-

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(a)

(b)

(c)

FIGURE 11.16

Comparison of the solution along the plume center-line in the ground level for

the (a) stable, (b) neutral, and (c) unstable boundary layers. The solid, dotted,

and dashed lines correspond to the solutions in the reference, telescopic, and

adaptive grids.

tances from the source. The adaptive mesh methods may also use fewer mesh points

than using fixed refined meshes since they are able to place mesh points where the

solution requires them rather than in pre-defined locations where they may not be nec-

essary for solution accuracy. However, there is an extra cost with the adaptive codes,

that of periodically refining/coarsening the mesh. In particular, the test cases have

demonstrated some important consequences of vertical mesh resolution for boundary-layer pollutant dispersion.

It is usual in tropospheric dispersion models to stretch the mesh in the vertical

domain and place more solution points near to the ground. Close to ground-level

sources, this often makes sense since it gives a better resolution of the initial stages

of vertical mixing and of deposition to the ground. However, at large distance from

their sources pollutants can become well mixed close to the ground and the important

feature is their escape from the boundary layer to higher levels of the troposphere.

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The results here demonstrate that for neutral and unstable boundary layers solution

accuracy requires refined meshes not close to the ground but close to the inversion

height where steep gradients can occur. The use of coarse meshes in this region could

have a significant affect on the prediction of pollutants mixing out of the boundarylayer for these conditions and may be a source of error in regional-scale, pollution-

dispersion models. In a realistic boundary-layer model, vertical mixing profiles will

change during the diurnal cycle making the a priori choice of vertical mesh structure

difficult. Adaptive refinement would seem to be the simplest method for resolving

such phenomena since the choice of mesh is made naturally according to the solution

structure resulting from different stability conditions.

Our general conclusion is that the adaptive MOL approach works well for two

space dimensional problems and in those cases it is possible to use standard codesproviding that it is possible to make use of sophisticated linear algebra methods that

are tailored to the problem. In the case of three-dimensional problems, however, it

seems more necessary to use tailor-made codes either based on the MOL as in [17]

or using the operator-splitting approach described here. Very recent work by Verwer

and others has suggested that the approach we used in two dimensions should also

be used in three space dimensions rather than introducing an operator-splitting error.

The challenge now is to implement this in a sufficiently efficient way to make the

MOL competitive with operator splitting in terms of efficiency.

Acknowledgments

This research has been supported by grants from NERC and from the Pakistan

Government for one of us (IA). The funding for the SPRINT2D and 3D software has

come from Shell Global Solutions. These calculations have been carried out on anOrigin2000 machine with support from a JREI grant. We would also like to thank 

our many colleagues who helped on this project such as G. Hart, J. Smith, M. Pilling,

and many others.

References

[1] I. Ahmad and M. Berzins, MOL solvers for hyperbolic PDEs with source terms,

submitted to special issue of Mathematics and Computers in Simulation.

[2] I. Ahmad and M. Berzins, An algorithm for ODEs from atmospheric dispersion

problems, Applied Numerical Mathematics, 25, (1997), 137–149.

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[3] M. Berzins, Temporal error control for convection-dominated equations in two

space dimensions, SIAM Journal on Scientific and Statistical Computing, 16,

(1995), 558–580.

[4] M. Berzins and J.M. Ware, Solving convection and convection reaction prob-

lems using the MOL, Applied Numerical Mathematics, 20, (1996), 83–99.

[5] M. Berzins, P.M. Dew, and R.M. Furzeland, Developing software for time-

dependent problems using the method of lines and differential algebraic inte-

grators, Applied Numerical Mathematics, 5, (1989), 375–390.

[6] M. Berzins and J.M. Ware, Positive cell-centered finite volume discretiza-

tion methods for hyperbolic equations on irregular meshes, Applied Numerical

 Mathematics, 16, (1995), 417–438.

[7] M. Berzins, R. Fairlie, S.V. Pennington, J.M. Ware, and L.E. Scales,

SPRINT2D: Adaptive Software for PDEs, ACM Transactions on Mathemat-

ical Software, 24, (1998), 475–499.

[8] T.J. Barth and D.C. Jesperson, The design and application of upwind schemes

on unstructured meshes, AIAA-89-0366, (1989), 9–12.

[9] T.J. Barth, Numerical aspects of computing viscous high reynolds number flowson unstructured meshes, AIAA Paper 91-0721, 29th Aerospace Sciences Meet-

ing, January 7–10, (1991), Reno Nevada.

[10] R. Biswas and R.C. Strawn, A new procedure for dynamic adaption of 3D

unstructured grids, Applied Numerical Mathematics, 13, (1994), 437–452.

[11] D.P. Chock, A comparison of numerical methods for solving the advection

equation III, Atmos. Env., 25A, (1991), 553–571.

[12] S. Ghorai, A.S. Tomlin, and M. Berzins, Resolution of pollutant concentrationsin the boundary layer using a fully 3Dadaptivegridding technique, Atmospheric

 Environment, 34, 18, (2000), 2851–2863.

[13] G. Hart, A.S. Tomlin, J. Smith, and M. Berzins, Multi-scale atmospheric dis-

persion modelling by use of adaptive gridding techniques, Environmental Mon-

itoring and Assessment, 52, (1998), 225–238.

[14] Ø. Hov, Z. Zlatev, R. Berkowicz, A. Eliassen, and L.P. Prahm, Comparisons of 

numerical techniques for use in air pollution models with non-linear chemicalreactions, Atmospheric Environment, 23, (1989), 967–983.

[15] H.J. Jacobs, H. Feldman, H. Kass, and M. Messesheimer, The use of nested

models for air pollution studies: an application of the EURAD model to a

SANA episode, Journal of Applied Meteorology, 34, (1995), 1301–1319.

[16] B. Joe and R.B. Simpson, Triangular meshes for regions of complicated shape,

 Int. J. Numer. Meth. Eng., 23, (1991), 987–997.

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[17] C.R. Johnson, M. Berzins, L. Zhukov, and R. Coffey, SCIRun: Application to

atmospheric dispersion problems using unstructured meshes, pp. 111–122 in

 Numerical Methods for Fluid Dynamics VI, M.J. Baines, ed., ICFD, Wolfson

Building, Parks Road, Oxford. ISBN 0 9524929 11, (1998).

[18] R.J. Leveque and H.C. Yee, A study of numerical methods for hyperbolic con-

servation laws with stiff source terms, Journal of Computational Physics, 86,

(1990), 187–210.

[19] S.C. Liu, M. Trainer, F.C. Fehsenfeld, D.D. Parrish, E.J. Williams, D.W. Fahey,

G. Hubler, and P.C. Murphey, Ozone production in the rural troposphere and

the implications for regional and global ozone distributions, J. Geophys. Res.,

92, (1987), 4191–4207.[20] R. Mathur and L.K. Peters, Adjustment of wind fields for application in air

pollution modelling, Atmospheric Environment, 24A, (1990), 1095–1106.

[21] S.V. Pennington and M. Berzins, New NAG library software for first-order

partial differential equations, ACM Transactions on Mathematical Software,

20, (1994), 63–99.

[22] L.K. Peters, C.M. Berkovitz, G.R. Carmichael, R.C. Easter, G. Fairweather, S.J.

Ghan, J.M. Hales, L.R. Leung, W.R. Pennell, F.A. Potra, R.D. Saylor, and T.T.Tsang, The current and future direction of Eulerian models in simulating the

tropospheric chemistry and transport of trace species: a review, Atmospheric

 Environment, 29, (1995), 189–222.

[23] M. Putti and C. Cordes, Finite element approximation of the diffusion operator

on tetrahedra, SIAM Journal on Scientific Computing, 19, (1998), 1154–1168.

[24] T. Rajaona, M.C. Ciccoli, and A. Coppalle, Local refinement around a pollu-

tion source using a domain decomposition method, in Proc. Int. Conf. on Air 

Pollution Modelling and Simulation, Paris, France (1998).

[25] J.H. Seinfeld, (1986) Air Pollution, Wiley, New York.

[26] S. Sillman, J.A. Logan, S.C. Wofsy, J. Geophys. Res., 95, (1990), 1837.

[27] W. Skamarock, J. Oliger, and R.L. Street, Adaptive grid refinement for numer-

ical weather prediction, Journal of Computational Physics, 80, (1989), 27–60.

[28] W. Speares and M. Berzins, A 3D unstructured mesh adaptation algorithm for

time-dependent shock-dominated problems, International Journal for Numer-ical Methods in Fluids, 25, (1997), 81–104.

[29] S. Spekreijse, Multigrid solution of monotone second order discretizations of 

hyperbolic conservation laws, Math. Comp., 47, 179, (1987), 135–155.

[30] S. Sunderam, J.A. Logan, and S.C. Wofsy, A regional scale model for ozone in

the united states with subgrid representation of urban and power plant plumes,

 Journal of Geophysical Research, 95, (1990), 5731–5748.

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[31] O. Talat, A quantitative analysis of numerical diffusion introduced by advection

algorithms in air quality models, Atmospheric Environment, 31, (1997), 1933–

1940.

[32] A.S. Tomlin, M. Berzins, J. Ware, J. Smith, and M.J. Pilling, On the use of 

adaptive gridding methods for modelling chemical transport from multi-scale

sources, Atmospheric Environment, 31, (1997), 2945–2959.

[33] A.S. Tomlin, S. Ghorai, G. Hart, and M. Berzins, The use of 3D adaptive un-

structured meshes in pollution modelling, in Large-Scale Computations in Air 

Pollution Modelling, Z. Zlatev et al., ed., (1999) Kluwer Academic Publishers.

[34] M. VanLoon, Numerical methods in smog prediction, Ph.D. Thesis, (1996)

CWI Amsterdam.

[35] I. Verwer, Gauss–Seidel iteration for stiff ODEs from chemical kinetics, SIAM 

 Journal on Scientific Computing, 15, (1994), 1243–1250.

[36] C.B. Vreugdenhil and B. Koren, Numerical methods for advection-diffusion

problems, Notes on Numerical Fluid Mechanics, 45, (1993), Vieweg, Braun-

schweig/Weisbaden, ISBN 3-528-07645-3.

[37] M. Wierse, A new theoretically motivated higher order upwind scheme on

unstructured grids of simplices, Advances in Computational Mathematics, 7,(1997), 303–335.

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Chapter 12

Two-Dimensional Model of a Reaction-Bonded Aluminum Oxide Cylinder

M.J. Watson, H.S. Caram, H.M. Chan, M.P. Harmer, Ph. Saucez,

A. Vande Wouwer, and W.E. Schiesser

12.1 Introduction

The reaction-bonded aluminum oxide (RBAO) process has been shown to have

many advantages over conventional ceramic processing [1, 2, 3, 4]. The RBAO

process starts with intensely milled aluminum and Al2O3 powders which can be

compacted into a variety of shapes and sizes. The main advantage of the process

is that the compacted shapes are strong enough to be machined prior to firing. The

porous, compacted samples are then heat treated in air, to oxidize the aluminum,

producing Al2O3-based ceramics.

The oxidation of aluminum is highly exothermic and, as a consequence, an ignitionfront has been observed passing over the sample’s surface during firing. An ignition

wavefront is undesirable for the RBAO process because both thermal and chemical

stresses are developed. The thermal stresses are transitory and are caused by the large

temperature difference between the hot reaction zone and the cooler unreacted zone.

The chemical stresses are caused by the 28% volumetric expansion associated with

the oxidation of aluminum. The chemical stresses are not transitory, but remain in the

wake of the ignition wave-front, as is evident from the steep composition gradients in

the radial direction, shown in Figure 12.1. The depth of the reacted shell is restricted

by the radial diffusion of oxygen through the pores of the rod. Sample failure isusually observed soon after ignition.

In this chapter a two-dimensional, transient model of the reaction behavior of a

RBAO rod is developed. The model can be used to investigate the reaction behavior

under a variety of experimental conditions, including variations in initial aluminum

concentration, furnace heating cycles, furnace atmosphere, and furnace temperature

gradients. Specifically, the model is used here to describe the reaction behavior

during ignition. The results from the model, which gives the oxygen concentration,

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FIGURE 12.1

Light optical micrograph of the cross-section of an RBAO rod after an ignition

front has passed.

aluminum concentration, and temperature as a function of position and time, can thenbe used to evaluate the stress distribution within the sample.

In this study we use the method of lines (MOL) to solve the set of three two-

dimensional simultaneous partial differential equations (PDEs), developed in the

next section. Three variations, based on three different configurations of the two-

dimensional spatial mesh, are used to calculate the spatial derivatives:

1. fixed, equally spaced nodes in the axial and radial directions

2. fixed, equally spaced nodes in the axial direction, and fixed, non-equally spaced

nodes in the radial direction

3. time-adapted nodes in the axial direction, and fixed, non-equally spaced nodes

in the radial direction

In Section 12.2 the three PDEs that are used to describe the distribution of temper-

ature, aluminum concentration, and oxygen concentration are developed. They are

presented in dimensionless form with the appropriate initial and boundary conditions.

The equations are solved using the numerical method of lines. The three variations

listed above are used to evaluate the spatial derivatives. In Section 12.3 the numerical

solutions are presented, and the different mesh configurations are compared. The

results are discussed in Section 12.4 and possibilities for improvement are given.

Partially reacted

transition

Unreacted core

Reacted shell

1mm

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12.2 Model Development

The following is the development of a continuum model that describes the reac-

tion behavior within a cylinder of RBAO. The model describes changes in oxygen

concentration, aluminum concentration, and sample temperature with time along the

axial and radial directions of the cylinder.

12.2.1 Model Assumptions

1. The bulk (furnace) concentration of oxygen is given by the ideal gas law:

CO2,∞ =(0.21)P 

RgT ∞,

where T ∞ is the furnace temperature, Rg is the gas constant, P  is the furnace

pressure (atmospheric), and 0.21 is the mol fraction of O2 in air.

2. For the reaction

2Al +3

2O2 −→ Al2O3 ,

the reaction rate, , is described by

= k(T )CO2C2

Al , (12.1)

where CAl is the concentration of aluminum, CO2 is the concentration of oxy-

gen, and k(T) is the reaction rate constant. The rate constant, k(T ), is described

by Arrhenius temperature dependence given by:

k(T ) = k0 exp−Ea

Rg

T  , (12.2)

where k0 is the pre-exponential factor, Ea is the activation energy, and T  is

absolute temperature. The expression for the reaction rate is empirical and is

the same as that used in a similar study [5].

3. There is no variation in the θ -direction. Only variations in the radial direction,

r , and the axial direction, z, are considered.

4. For simplicity, properties such as specific heat, density, diffusivity, and con-

ductivity are kept constant.

12.2.2 Continuum Model Equations

The aluminum in the rod reacts to form Al2O3. The general mass balance for

aluminum is given by:

∂CAl

∂t = −2 , (12.3)

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where t  is time and the coefficient of 2 is required for stoichiometry.

The oxygen balance is given by:

∂CO2

∂t = D∇ 2CO2 − 3

2 , (12.4)

where D is the effective diffusivity of oxygen in the pore structure, ∇ 2 is the Lapla-

cian operator, and the coefficient of 3/2 is required for stoichiometry. Ignoring any

variation in the θ -direction, Equation 12.4 becomes:

∂CO2

∂t = D

∂2CO2

∂z2+

1

r

∂

∂rr

∂CO2

∂r

−

3

2 . (12.5)

The energy balance is given by:

ρcp∂T 

∂t = λ∇ 2T  + (−H ) , (12.6)

where ρ is the density, cp is the specific heat, λ is the thermal conductivity, and

(−H ) is the heatofreaction. Ignoringany variationin the θ -direction,Equation12.6

becomes:

∂T ∂t 

= α

∂

2

T ∂z2 + 1

r∂

∂rr ∂T 

∂r

+ (−H )

ρcp , (12.7)

where α = kρcp

is the thermal diffusivity.

12.2.3 Initial and Boundary Conditions

The initial conditions, at t  = 0, for the cylinder are

CAl (0, z , r ) = CAl,0

T (0, z , r ) = T 0

CO2(0, z , r ) = CO2,0 =

0.21P 

RgT 0, (12.8)

where the initial concentration of aluminum and initial temperature are constants and

independent of r and z. The initial oxygen concentration is found from the ideal gas

law.

Heat is transferred to or from the furnace to the outer surface of the cylinder throughconvective and radiative heat transfer. A flux balance on the left- and right-hand sides

of the cylinder, corresponding to z = 0 and z = L, respectively, gives the boundary

conditions for the energy balance:

λ∂T 

∂z(t, 0, r) = h (T (t , 0, r) − T ∞) + σ

T 4(t, 0, r) − T 4∞

λ∂T 

∂z(t , L , r) = −h (T ( t , L , r ) − T ∞) − σ

T 4(t , L , r) − T 4∞

, (12.9)

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where T ∞ = T ∞(t,z,r) is the furnace temperature, h is the heat-transfer coefficient,

σ  is the Stefan–Boltzman constant, and is the emmisivity. A heat-flux balance on

the curved surface of the cylinder, corresponding to r = R, gives:

λ∂T 

∂r(t,z,R) = −h (T (t , z , R ) − T ∞) − σ

T  4(t,z,R) − T  4∞

, (12.10)

while the symmetry condition at the center of the cylinder, r = 0, gives:

∂T 

∂r(t,z, 0) = 0 . (12.11)

Balancing the mass flux of oxygen at the left, z = 0, and right, z = L, boundaries

of the cylinder yields:

D∂CO2

∂z(t , 0, r) = km

CO2

(t, 0, r) − CO2,∞

D∂CO2

∂z(t , L , r) = −km

CO2 (t , L , r) − CO2,∞

, (12.12)

where km is the mass transfer coefficient and C∞ is the concentration of oxygen in the

furnace. Balancing the mass flux on the curved surface of the cylinder, correspondingto r = R, gives:

D∂CO2

∂r(t,z,R) = −km

CO2

(t,z,R) − CO2,∞

, (12.13)

while the symmetry condition at the center of the cylinder, r = 0, gives:

∂CO2

∂r

(t,z, 0) = 0 . (12.14)

12.2.4 Parameters

The model parameters that are used are listed in Table 12.1. The kinetic parameters,

Ea and k0, are chosentoagreewithanexistingmodel thatdescribes the RBAO reaction

behavior of a flat slab [5]. The other parameters are either measured or calculated

from correlations found in the literature.

12.2.5 Dimensionless Equations

The set of PDEs can be rendered dimensionless by using the following variables:

u =CAl

CAl,0, v =

CO2

CO2,0, w =

T 

T ad 

,

ξ  =r

R, η =

z

L, τ  =

k0 exp(−γ )CAl,0CO2,0

t ,

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Table 12.1 Model Parameters

Property Symbol Value Unit

Length L 0.1 mRadius R 0.002 m

Density ρ 2460 kg/m3

Specific heat cp 1000 J/(kg K)

Conductivity λ 1.4 W/(m K)

Heat of formation (−H ) 1669792 J/mol

Gas constant Rg 8.314 J/(mol K)

Activation energy Ea 170000 J/mol

Arrhenius factor k0 300 m6 /(mol2s)

Heat-transfer coefficient h 15 W/(m2K)Stefan-Boltzman constant σ  5.67e-8 W/(m2K4)

Emissivity 0.1

Diffusivity D 1.5×10−6 m2 /s

Mass-transfer coefficient km 0.03 m/s

Initial Al concentration CAl,0 20000 mol/m3

Initial temperature T 0 750 K

where

T ad  = T 0 +(−H)CAl,0

2ρCp

, γ  =Ea

RgT ad 

, (12.15)

and T ad  represents the adiabatic temperature rise. The PDEs become:

∂u

∂τ = −2 exp

γ 

1 −

1

w

vu2 (12.16)

∂v∂τ 

= −ψ exp

γ 

1 − 1w

vu2 + χR

1ξ 

∂∂ξ 

ξ ∂v∂ξ 

+ χL ∂

2

v∂η2

(12.17)

∂w

∂τ = +β exp

γ 

1 −

1

w

vu2 + φR

1

ξ 

∂

∂ξ ξ 

∂w

∂ξ 

+ φL

∂2w

∂η2(12.18)

where

φL =α exp(γ )

L2k0CAl,0CO2,0, φR =

α exp(γ )

R2k0CAl,0CO2,0, β =

(−H)CAl,0

ρCpT ad 

,

χL =D exp(γ )

L2k0CAl,0CO2,0, χR =

D exp(γ )

R2k0CAl,0CO2,0, ψ =

3CAl,0

2CO2,0.

In dimensionless variables, the initial conditions become:

u(0, η , ξ ) = 1

v(0, η , ξ ) = 1

w(0, η , ξ ) = 1 −β

2. (12.19)

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The thermal boundary conditions become:

∂w

∂η(τ, 0, ξ ) = κL(w − w∞) + ϒ L w4 − w4

∞∂w

∂η(τ, 1, ξ ) = −κL(w − w∞) − ϒ L

w4 − w4

∞

∂w

∂ξ (τ,η, 1) = −κR(w − w∞) − ϒ R

w4 − w4

∞

∂w

∂ξ (τ,η, 0) = 0 , (12.20)

where

κL =Lh

λ, ϒ L =

σLT 3ad 

λ, κR =

Rh

λ, ϒ R =

σRT 3ad 

λ,

and the boundary conditions for oxygen mass transfer become:

∂v

∂η(τ, 0, ξ ) = L(v − v∞)

∂v∂η

(τ, 1, ξ ) = −L(v − v∞)

∂v

∂ξ (τ,η, 1) = −R(v − v∞)

∂v

∂ξ (τ,η, 0) = 0 , (12.21)

where

L = LkmD

, R = RkmD

.

12.2.6 Method of Solution

By dividing the axial coordinate, η, into N L spatial points, and the radial coor-

dinate, ξ  into N R spatial points, and using discrete approximations to describe the

spatial derivatives, the three PDEs are approximated by 3 × N L × N R ordinary

differential equations (ODEs). This is known as the numerical method of lines [6]

and is used to solve the simultaneous mass and energy balances with the appropriateboundary conditions. The set of ODEs is solved using the LSODES integrator [7, 8, 9]

for stiff ODEs with a sparse Jacobian matrix. The integrator uses an implicit method,

based on backward differentiation formulas, of order 5 with step-size control. The

maximum absolute and relative integration errors are set to 10−5. The parameters

that are used to solve the equations are listed in Table 12.2.

Three variations are used to calculate the distribution of the nodes and the spatial

derivatives.

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Table 12.2  Solution Parameters

Parameter Symbol Value

Axial nodes NL 26Radial nodes NR 11

Tolerance 10−5

Adaptation parameter α 1

Adaptation parameter β 100

Method 1. The first variation uses fixed, equally spaced nodes in the radial and axial

directions. The second-order spatial derivatives are calculated using subrou-tine DSS044 and the first-order spatial derivatives in the radial direction are

calculated using subroutine DSS034 from the DSS/2 package [10].

Method 2. The second variation uses fixed, equally spaced nodes in the axial direc-

tion, and fixed, non-equally spaced nodes in the radial direction, using sub-

routine DSS044 to calculate the second-order spatial derivatives in the axial

direction, and subroutine DSS032 to calculate the first- and second-order spa-

tial derivatives in the radial direction.

Method 3. The third variation uses fixed, non-equally spaced nodes in the radial

direction, and time-adapted nodes in the axial direction. SubroutineAGE [11] is

used to adapt the axial nodes. Finite differences, as implemented in subroutine

 WEIGHTS [12] are used to calculate the second-order spatial derivative in the

axial direction. Subroutine DSS032 is used to calculate the first- and second-

order spatial derivatives in the radial direction.

These three methods will be referred to as Methods 1, 2, and 3, respectively,

throughout.

The spatial remeshing algorithm, AGE, has been applied to a variety of one-

dimensional transient problems [11]. Here it is applied to the two-dimensional prob-

lem [Equations (12.16) through (12.21)] by basing the grid adaptation in the axial

direction on the solution at r = R, or ξ  = 1. The spatial remeshing algorithm is

based on the second derivative:

 ηk+1

i

ηk+1i−1

α +

∂2U 

∂η2 (t k+1, η)

∞

dη ≈ constant , (12.22)

where U(t,η) = {u(t, η, 1),v(t,η, 1),w(t,η, 1)}. A parameter β is used to avoid

excessive grid distortion, i.e., the value of the derivatives that exceed β are reduced

to β. Values of α and β are given in Table 12.2. The grid adaptation is performed at

each print time interval (i.e., in contrast with the static gridding methods presented in

Chapter 1, grid adaptation occurs at regular time intervals rather than after a certain

number of variable time steps).

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12.3 Results

In this section the numerical solutions of Equations (12.16) through (12.21) are

presented. The three different configurations of the two-dimensional spatial mesh

are used to evaluate the spatial derivatives and the different numerical solutions are

compared.

12.3.1 Furnace Conditions

To simulate the firing of a sample in a furnace, the sample is heated at 5 K/min and

a linear temperature gradient of 40 K is assumed from the left end of the sample to

the right end. The initial furnace temperature is 750 K (= T 0), thus:

T ∞(t,z) = T 0 +5

60t  − 40

z

L, (12.23)

and in dimensionless units:

w∞(τ,η) = 1 −β

2+ 

5τ 

60T ad 

k0

exp(−γ )CAl,0

CO2,0

−40η

T ad 

. (12.24)

Integration continues for 14 min, until the furnace temperature reaches 820 K. The

temperature and concentrations are recorded 100 times within this period.

12.3.2 Numerical Solutions

Figures 12.2, 12.3, and 12.4 show the time progression of the temperature and

concentration distributions of the ignited rod. The shading of the solution has been

interpolated for clarity. The high temperature ignition front is seen propagating fromthe hot end on the left to the right of the sample in Figure 12.2. Similarly in

Figure 12.3, a region of reacted aluminum (black) follows the ignition front. The

depth of the reacted zone is limited by the rate of diffusion of oxygen into the sample

relative to the rate of reaction. This is shown in Figure 12.4. All of the oxygen within

the porous structure of the sample is consumed by the oxidation reaction. As fresh

oxygen from the furnace diffuses into the rod, it is consumed near the surface before

it is allowed to diffuse to the center. The limited depth of the reaction zone agrees

qualitatively with the micrograph of  Figure 12.1.

A comparison of the node placement of methods 1, 2, and 3, at t  = 210 sec, is

shown in Figure 12.5. The shading of the space between the nodes is based on the

value of the upper left node. Figure 12.5(a) shows the solution with equally spaced

nodes in the radial and axial direction. Figures 12.5(b) and (c) have radial grid spacing

based on concentric cylinders of equal volume, V :

V  = π r2i − π r2

i−1 =π R2

N R − 1. (12.25)

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Axial position (mm)

   R  a   d   i  a   l  p  o  s   i   t   i  o  n   (  m  m   )

   T  e  m  p  e  r  a   t  u  r  e   (   K   )

(a)

(b)

(c)

800

900

1000

1100

1200

1300

1400

1500

2

1

0

2

1

0

2

1

00 10 20 30 40 50 60 70 80 90 100

FIGURE 12.2

Temperature distribution within a RBAO rod: (a) 160 sec; (b) 210 sec; (c) 260

sec.

0 10 20 30 40 50 60 70 80 90 100

2

1

0

2

1

0

2

1

0

Axial position (mm)

   R

  a   d   i  a   l  p  o  s   i   t   i  o  n   (  m  m   )

(a)

(b)

(c)   A   l  u  m   i  n  u  m   c

  o  n  c  e  n   t  r  a   t   i  o  n   (   k  m  o   l   /  m   3   )

2

4

6

8

10

12

14

16

18

20

FIGURE 12.3

Aluminum concentration distribution within a RBAO rod: (a) 160 sec; (b) 210

sec; (c) 260 sec.

This allows for greater spatial resolution and accuracy close to the surface of the

cylinder, where the majority of the oxidation reaction occurs during ignition. Fig-

ure 12.5(c) shows the solution when axial grid adaptation is used. The grid points

are spaced closer together near the left end, and at the ignition front near the center.

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0 10 20 30 40 50 60 70 80 90 100

2

1

0

2

1

0

2

1

0

Axial position (mm)

   R  a   d   i  a   l  p  o  s   i   t   i  o  n   (  m  m   )

(a)

(b)

(c)   O  x  y  g  e  n

  c  o  n  c  e  n   t  r  a   t   i  o  n   (   k  m  o   l   /  m   3   )

0

0.5

1

1.5

2

2.5

3

FIGURE 12.4

Oxygen concentration distribution within a RBAO rod: (a) 160 sec; (b) 210 sec;

(c) 260 sec.

0 10 20 30 40 50 60 70 80 90 100

1

1.0

2

0.5

0

1

1.0

2

0.5

0

1

1.0

2

0.5

0

Axial position (mm)

   R  a   d   i  a   l  p  o  s   i   t   i  o

  n   (  m  m   )

(a)

(b)

(c)

   A   l  u  m   i  n  u  m   c

  o  n  c  e  n   t  r  a

   t   i  o  n   (   k  m  o   l   /  m   3   )

2

0

4

6

8

10

12

14

16

18

20

FIGURE 12.5

Aluminum concentration distribution at t  = 210 sec showing a comparison of 

the node placements. The nodes are distributed according to: (a) Method 1;

(b) Method 2; (c) Method 3.

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The close spacing of the grid points follows the regions where concentration and

temperature gradients are steep.

Figures 12.6 and 12.7 show the temperature and concentration distributions, re-

spectively, on the outer surface of the rod, corresponding to r = R. The nodes aredistributed evenly in the axial direction in Figures 12.6(a) and (b) and 12.7(a) and

800

1200

1600

800

1200

1600

800

1200

1600

0 10 20 30 40 50 60 70 80 90 100

Axial position (mm)

   T  e  m  p  e  r  a   t  u  r  e   (   K   )

(a)

(b)

(c)

FIGURE 12.6

Axial temperature distribution at r = R at t  = 160, 185, 210, 225, and 260

sec. The nodes are distributed according to: (a) Method 1; (b) Method 2;

(c) Method 3.

0

10

20

0

10

20

0 10 20 30 40 50 60 70 80 90 1000

10

20

Axial position (mm)

(a)

(b)

(c)

   A   l  u  m   i  n  u  m   c

  o  n  c  e  n   t  r  a   t   i  o  n   (   k  m  o   l   /  m   3   )

FIGURE 12.7

Axial concentration distribution at r = R at t  = 160, 185, 210, 225, and 260

sec. The nodes are distributed according to: (a) Method 1; (b) Method 2;

(c) Method 3.

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to 11 kmol/m3 at the next node in. On the grid spaced on equal volumes, the zone of 

complete oxidation, corresponding to a concentration of 0 kmol/m3 has penetrated

0.1 mm into the sample.

Both solutions show a small region where the concentration goes beyond the initialconcentration. This suggests that more grid points are required in the radial direction

to improve the accuracy of the solution.

12.4 DiscussionA banded diagonal structure of the ODEs, and hence the Jacobian matrix, cannot be

maintained for this two-dimensional problem. The large size of the Jacobian, caused

by the large number of ODEs, excludes the use of full Jacobian matrix evaluation.

The integrator LSODES was chosen because of the efficient nature of the sparse

Jacobian matrix calculation. This allowed for fast computation of the solution to the

problem when the grid was fixed, as in Methods 1 and 2, where the calculation took 

2.8 min and 3.2 min, respectively, on a 333 MHz PC. The solution took considerably

longer (14.5 min) to calculate when Method 3 was used. This is because LSODESuses variable order backward differentiation formulas to perform each integration

step. Every time the grid was adapted the integrator had to re-initialize, which is

computationally expensive. Ideally, a single-step solver, such as the implicit Runge–

Kutta integratator RADAU5 [13], should be used.

Subroutine AGE successfully tracked the ignition wavefront and adapted the grid

points in the region of the wavefront. The spatial grid was adapted at specified discrete

time intervals, determined by the printing time interval. Perhaps greater success could

be achieved by using the adaptation algorithm after a fixed number of integrator time

steps. This would allow the integrator, which has time step-size control, to adapt

more frequently when the solution to the problem is changing faster.

For comparison purposes, results were obtained with the adaptive mesh refinement

algorithm AGEREG discussed in Chapter 2, i.e., using a time-varying number of nodes

adapted periodically after a certain number of steps, and the RK solver RADAU5.

The dimensionless oxygen concentration distribution is shown at 14 equally spaced

time intervals at r = R in Figure 12.9. The results were compared with 111 fixed,

equally spaced nodes, shown in Figure 12.10. The number of fixed nodes was chosen

so as to have the same CPU time as with a variable number of nodes computed by

AGEREG. The computational statistics are summarized in Table 12.3, which compares

the number of axial nodes NL, the number of function evaluations FNS, the number of 

Jacobian evaluations JACS, and the number of computed steps STEPS. The evolution

of the number of grid points used by AGEREG was 101, 48, 49, 50, 52, 87, 93, 102,

95, 81, 67, 67, 55, 51, 52, 51 for the 14 time intervals shown in Figure 12.9. The

results give similar global accuracy, but improved resolution of small-scale solution

features near the boundaries at z = 0 and 1 when adaptive grids are used.

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FIGURE 12.9

Dimensionless oxygen concentration distribution at ξ  = 1 using a variable num-

ber of nodes.

FIGURE 12.10

Dimensionless oxygen concentration distribution at ξ  = 1 using 111 fixed,

equally spaced nodes.

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Table 12.3 Computational Statistics

Grid NL FNS JACS STEPS

AGEREG 48 – 101 18229 887 1829uniform 111 13637 662 1333

The results obtained thus far suggest that further improvements could probably be

achieved by using 2D adaptation techniques (in the radial direction as well as the

axial direction) such as the one reported by Steinebach and Rentrop in Chapter 6.

12.5 Summary

A method to use a one-dimensional spatial remeshing algorithm on a two-dimen-

sional problem, in cylindrical coordinates, was presented. The axial grid adaptation

was based on the condition of the second derivative at the outer surface of the cylinder.

The solution displayed a moving ignition wave-front. The algorithm was able to

focus the axial nodes in the region of the ignition front, thereby improving the spatialresolution and accuracy in that region. By appropriately choosing the numerical

integrator, similar global accuracy was obtained using a fixed, equally spaced grid,

but better local accuracy was obtained on an adapted grid with a variable number of 

nodes, without compromising computational efficiency.

Acknowledgment

I gratefully acknowledge the kind financial support of the U.S. Office of Naval

Research (grant N0014-96-1-0426) under the monitoring of Dr. S. Fishman.

References

[1] N. Claussen, R. Janssen, and D. Holz, The Reaction Bonding of Aluminum

Oxide, Journal of the Ceramic Society of Japan, 103(8), (1995), 1–10.

[2] N. Claussen, S. Wu, and D. Holz, Reaction Bonding of Aluminum Oxide

(RBAO) Composites: Processing, Reaction Mechanisms and Properties, Jour-

nal of the European Ceramic Society, 14, (1994), 97–109.

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[3] D. Holz, S. Wu, S. Scheppokat, and N. Claussen, Effect of Processing Parame-

ters on Phase and Microstructure Evolution in RBAO Ceramics, Journal of the

 American Ceramic Society, 77(10), (1994), 2509–2517,

[4] S. Wu, D. Holz, and N. Claussen, Mechanisms and Kinetics of Reaction–

Bonded Aluminum Oxide Ceramics, Journal of the American Ceramic Society,

76(4), April 1993, 970–980.

[5] S.P. Gaus, M.P. Harmer, H.M. Chan, and H.S. Caram, Controlled Firing of 

Reaction-Bonded Aluminum Oxide (RBAO) Ceramics: Part I, Continuum

Model Predictions, Journal of the American Ceramic Society, 82(4), April

1999, 897–908,

[6] W.E. Schiesser, The Numerical Method of Lines: Integration of Partial Differ-ential Equations, Academic Press, San Diego, 1991.

[7] A.C. Hindmarsh, LSODE and LSODI, Two Initial Value Ordinary Differential

Equation Solvers, ACM — Signum Newsletter, 15(5), (1980), 10–11.

[8] S.C. Eisenstat, M.C. Gursky, M.H. Schultz, and A.H. Sherman, Yale Sparse

Matrix Package: I. The Symmetric Codes, Technical Report 112, Department

of Computer Sciences, Yale University, 1977.

[9] S.C. Eisenstat, M.C. Gurskey, M.H. Schultz, and A.H. Sherman, Yale SparseMatrix Package: II. The Nonsymmetric Codes, Technical Report 114, Depart-

ment of Computer Sciences, Yale University, 1977.

[10] W.E. Schiesser, DSS/2 (Differential Systems Simulator, version 2.) An Intro-

duction to the Numerical Method of Lines Integration of Partial Differential

 Equations, Lehigh University, Bethlehem, PA, 1977.

[11] P. Saucez, A. Vande Wouwer, and W.E. Schiesser, Some Observations on a

StaticSpatial RemeshingMethodBased on Equidistribution Principles, Journalof Computation Physics, 128, (1996), 274–288.

[12] B. Fornberg, Generation of Finite Difference Formulas on Arbitrarily Spaced

Grid, Mathematics of Computation, 51(184), (1988), 699–706.

[13] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and 

 Differential-Algebraic Problems, Springer Series in Computation Mathematics

14, Springer-Verlag, Berlin, 1991.

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Chapter 13

 Method of Lines within the Simulation Environment Diva for Chemical Processes

R. Köhler, K.D. Mohl, H. Schramm, M. Zeitz,A. Kienle, M. Mangold, E. Stein, and E.D. Gilles

13.1 Introduction

In chemical engineering, detailed process modeling, simulation, nonlinear analy-

sis, and optimization of single process units as well as integrated production plantsare issues of growing importance. As a software tool addressing these issues, the sim-

ulation environment Diva [17, 26] is introduced. Diva comprises tools for process

modeling, preprocessing, and code generation of simulation models. Furthermore, its

simulation kernel contains several advanced methods for simulation, parameter con-

tinuation, and dynamic optimization which can be applied to the same process model.

These numerical methods require a model description in a form of differential alge-

braic equations (DAE). However, many models of chemical processes derived from

first principles lead to partial differential equations (PDE) for distributed parameter

systems, to integro partial differential equations (IPDE) for population balances of 

dispersed phases, and to DAE for lumped parameter systems.

In order to transform the PDE and IPDE models into the required DAE model

formulation, Diva employs the “method-of-lines” (MOL) approach for one space

coordinate. The wide variety of distributed parameter models of chemical processes

requires on one hand conventional discretization methods like, e.g., finite-difference

schemes, and on the other hand more sophisticated methods to obtain reliable results

in an acceptable computation time. One common feature of all advanced methods is

the use of some type of adaptive strategy. In moving-grid methods, the adaptationconcerns the positions of the grid points in the discretized spatial domain. Another

approach of so-called high-resolution methods uses adaptive approximation polyno-

mials. High-resolution methods are developed for hyperbolic conservation laws with

steep moving fronts. Examples are essentially non-oscillatory (ENO) schemes [33]

or the robust upwind κ-interpolation scheme [16].

In order to support the user of the simulation environment Diva, the symbolic

preprocessing tool SyPProT for MOL discretization is developed by means of the

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computer-algebra-system Mathematica. This tool provides different discretiza-

tion schemes to transform PDE, IPDE, and the related boundary conditions on 1-

dimensional spatial domains into DAE. Discretization options concern fixed spatial

grids and moving grids based on an equidistribution principle [43]. The toolboxarchitecture of SyPProT allows fast testing of various discretization schemes with-

out redefinition of the model equations. This is regarded as an important feature

to minimize the overall effort for modeling and simulation. A further characteristic

that is key to the easy exchange of discretization schemes is the separation between

model equations and MOL parameter definitions by means of theMathematica data

structure (MDS). The resulting overall DAE are written in symbolic form as an input

file for the Diva code generator [15, 31], which automatically generates the Diva

simulation files representing a process unit model in the model library.In the following section, the architecture of the simulation environment Diva is

presented. This architecture consists of four layers, i.e., the Diva simulation ker-

nel, the code generator, the symbolic preprocessing tool SyPProT, and the process

modeling tool ProMoT. The MOL discretization of PDE and IPDE as the main pre-

processing feature is the focus of Section 13.3. Standard discretization schemes like

finite-difference and finite-volume schemes as well as adaptive approaches like high-

resolution methods and an equidistribution principle based moving grid method are

explained for distributed parameter models with one space coordinate. The applica-

tion of these discretization methods within the symbolic preprocessing tool SyPProT

follows in Section 13.4, where the PDE and IPDE model representation as well as

the implemented MOL discretization capabilities are described. In Section 13.5, the

utilization of SyPProT and Diva is illustrated by simulations of a circulation-loop-

reactor model and a moving-bed chromatographic process model.

13.2 Architecture of the Simulation Environment Diva

The Diva architecture (Figure 13.1) comprises four layers that are all accessible

for editing and debugging by the user. The first or bottom layer contains the Diva

simulation kernel with the numerical methods and the library of generic process unit

models. The simulation kernel as well as the model representation are implemented in

Fortran77. A process unit model consists of severalFortran subroutines collected

in the model library and a data file for the parameters and initial values. A moredetailed description of the structure of the Diva simulation kernel, the linear implicit

DAE model representation, and its numerical methods follows in the next subsections.

The second layer of Diva consistsof the codegenerator (CG) and the corresponding

CG input file which uses the same standardized DAE representation as the simulation

kernel itself. The code generator automatically generates the Fortran subroutines

for a generic process unit model along with a data file for its parameterization, as

required by Diva.

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FIGURE 13.1

Architecture of the simulation environment Diva with the process modeling tool

ProMoT, the symbolic preprocessing tool SyPProT, the code generator, and

the Diva simulation kernel including the model library and the numerical DAE

methods [41, 40, 15, 17, 26, 31].

The third layer represents the symbolic preprocessing tool SyPProT as well as the

appropriate model definition file. The task of symbolic preprocessing concerns the

transformation of models derived from first principles into the DAE representation of 

Diva. The input format for representation of mixed DAE, PDE, and IPDE models is

theMathematica data structure (MDS).

The top layer of Diva is the process modeling tool ProMoT.

13.2.1 The Diva Simulation Kernel

13.2.1.1 Structure of the Diva Simulation Kernel

The Diva simulation kernel is a numerical tool that has been developed at the

University of Stuttgart over the last 15 years [11, 17, 10, 13, 26, 22]. It offers a wide

variety of numerical methods for the treatment of both process design and process

operation problems. The simulation kernel is organized in a modular structure as

depicted in Figure 13.2. Within the kernel, the user has the possibility to switch

interactively between numerical methods (second layer) using a command language,

which is handled by the command interpreter (top layer). All numerical methods can

be applied to the Diva plant model (third layer). This is possible due to the modular

structure of the Diva simulation kernel and the uniform representation of the unit

models (bottom layer, left). Using the plant flowsheet information, the individual

unit models are combined by the plant model processor to form the overall plant

model. The model representation will be discussed in more detail in the following

section.

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User

Plant Model

Unit ModelLibrary of

Processorcorrelations

physical property

Flowsheetinformation

Physical Property

parametersNumeric

parameters

Command Interpreter

Numerical Methods

Initial values

Unit Models

Code Library ofUnit

Models

Continuation &integrators Stability Analysis

OptimizationTools

NL-Equation-Handling

DAE-Analysis

Indexsolver

Event

DIVA Simulation Kernel

Generator

Unit Model parameters Simulationresults

Linearizedplant model

MATLAB

FIGURE 13.2

Structure of the Diva simulation kernel [17, 26].

13.2.1.2 DivaModel Representation

Within a flowsheet simulation tool, a process plant is represented as a structure of 

interconnected process unit models. Therefore, each process unit has to be specified

by a unit model. The plant model is then generated by connecting the unit models ac-

cording to the flowsheet of the process. To make connecting unit models easy, all unit

models have to be specified in a uniform way. In dynamic flowsheet simulation, the

formulation of unit models as systems of ordinary differential and algebraic equations

(DAE) is common practice for chemical engineering problems. For the simulationenvironment Diva, the formulation as semiimplicit DAE with a differential index of 

one has been chosen [42]. The model formulation of a process unit k is given as

follows:

Bk(xk, uk, pk, t) ·dxk

dt = f k(xk, uk, pk, t) t > t  0, xk(t 0) = x0

k , (13.1)

yk = H k · xk

with

process unit model index k ∈ I left side matrix Bk ∈ Rn×n

states xk ∈ Rn function vector f k ∈ Rn

inputs uk ∈ Rm outputs yk ∈ Rr

parameters pk ∈ Rp output matrix H k ∈ Rr×n

time t  ∈ R1 initial values x0k ∈ Rn

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The connections among the unit models are specified in a flowsheet information

file which contains a coupling matrix C = {Cij } that connects the internal outputs yj 

of the process unit j  to the internal inputs ui of the process unit i according to

ui = Cij  · yj , Cij  ∈ Rmi ×rj  . (13.2)

The plant model that is obtained by connecting the unit models is also a system of 

semi-implicit DAE. The numerical DAE methods of Diva are based on this uniform

DAE representation, into which PDE and IPDE have to be transformed by means of 

the MOL approach.

13.2.1.3 Sparse Numerical Methods within the Diva Simulation Kernel

Models of process plants typically result in highly nonlinear large sparse systems

of arbitrary structure. To keep computation times small, sparse matrix numerical

techniques are used within Diva. If sparse matrix techniques are used for the numer-

ical solution of (13.1), the patterns of the Jacobians ∂(Bk · xk)/∂xk , ∂(Bk · xk)/∂uk ,

∂f k/∂xk , ∂f k/∂uk have to be provided.

All numerical methods available in Diva can be applied to the same plant model.

The possibility of processing one plant model with different numerical methods

strongly reduces model implementation efforts. This is particularly importantbecausethe modeling and model implementation steps are still the most time-consuming steps

in computer-aided process engineering.

Besidesnumericalmethods provided byDiva, the export functions toMATLAB5.3

(a commercial software tool for matrix-based systems analysis and synthesis) offer

additional numerical and graphical capabilities. It is possible to generate a linearized

plant model in MATLAB format that can be used to apply control analysis and design

methods within MATLAB (Figure 13.2). Furthermore, MATLAB is used for the

visualization of simulation results.

Steady-State and Dynamic Simulation

Diva offers numerical methods for several different simulation tasks. For the so-

lution of initial value problems, several DAE integrators are available: SDASSL [2]

and SDASAC [3], using a backward differentiation method, LIMEXS using an ex-

trapolation method [4] and RADAU5S [7], using Runge–Kutta methods. A special

feature of SDASAC is the simultaneous calculation of parameter sensitivities (deriva-

tives of model states with respect to model parameters), which are needed, e.g., for

the solution of optimization and identification problems.Diva also provides different methods for solving systems of nonlinear algebraic

equations which are needed for the determination of steady-state solutions and the

consistent initialization of DAE integration. The solvers implemented in Diva are a

Levenberg-Marquardt algorithm [9] and a Newton method [28]. For the solution of 

steady-state equations there is also the possibility to calculate parameter sensitivities.

In addition to these standard numerical methods, Diva provides routines for event

handling. This includes explicit events caused by discrete input functions that are

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triggered at given times, and implicit events which are caused, e.g., by bounded

outputs of controllers and which depend on model states.

Continuation Methods for Nonlinear Analysis

Nonlinear effects have a strong influence on many chemical engineering processes,

especially in reaction engineering. When operating conditions are varied, nonlinear-

ities may cause sudden and often unexpected changes in the qualitative behavior of 

a process. Examples are transitions from a steady-state set point to a completely

different steady state with undesired properties or to oscillatory behavior [39]. The

knowledge of operating conditions under which such changes in the qualitative be-

havior occur are crucial for a safe and efficient operation of many chemical processes.

Continuation methods in combination with stability analysis have proven to be effi-cient tools for determining those boundaries between regions of qualitatively differ-

ent behavior. In Diva, continuation algorithms are available for the application to

all kinds of plant models. The methods comprise algorithms for the one-parameter

continuation of steady states as well as periodic solutions and algorithms for the two-

parameter continuation of saddle-node and Hopf bifurcations. The numerics have

been tailored to systems of high dynamical order. Within reasonable computation

time, they can perform a bifurcation analysis of differential algebraic systems from

several hundred to a few thousand equations as they typically result from a spatial

discretization by the MOL. In contrast, classical packages for bifurcation analysis

are restricted to systems of a few ordinary differential equations or to systems with

special structural properties. A detailed description of the numerical methods used in

Diva is beyond the scope of this chapter. The interested reader is referred to [13, 24].

In the following, only a brief idea of the capabilities of the methods will be given.

In general, a continuation algorithm can be used to trace the solution curve of an

under-determined system of algebraic equations

g(η) = 0, g ∈ Rm

, η ∈ Rm+1

(13.3)

in an (m + 1)-dimensional space. For that purpose a predictor-corrector algorithm

with local parameterization and step-size control is used in Diva. This algorithm is

able to cope with turning points, where the solution curve changes its direction.

A simple application of continuation methods is the computation of steady-state

solutions as a function of some distinguished model parameter p. In this case, g

is the right-hand side vector f  of the model equations, and η consists of the state

vector x and the parameter p. The result of such a one-parameter continuation of 

steady states can be visualized in a bifurcation diagram as shown in Figure 13.3. In

this example, the steady-state solutions form an S-shaped curve. A co-existence of 

three steady states is found for certain values of the bifurcation parameter F . An

eigenvalue monitor is used to determine the stability of the computed steady states

and to detect singularities, where one or several eigenvalues cross the imaginary axis.

In Figure 13.3, two types of steady-state singularities are found. The first type is

a so-called saddle-node bifurcation where two steady-state solutions coincide. The

second type is a Hopf bifurcation where a steady-state solution loses its stability and

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FIGURE 13.3Bifurcation diagram of a CSTR model: dependence of reactor temperature T 

on reactant flow F  [25].

gives rise to a branch of periodic solutions. In the diagram, the periodic solutions are

symbolized by the maximum temperature during a periodic cycle. From bifurcation

theory, necessary and sufficient conditions for saddle-node bifurcations and Hopf 

bifurcation can be derived. Together with the steady-state equations of the model,

they form augmented equation systems for the direct computation of the state vector

x and the parameter p at the singular points. Those augmented equation systems are

generated automatically by Diva. In the framework of the continuation algorithm,

they are used to trace the curves of singular points in two parameters. The resulting

curves form the boundary of the regions of qualitatively different behavior in the

parameter space. The predictor corrector algorithm in Diva is not only used for

the computation of steady-state solutions but also for the continuation of stable and

unstable periodic solutions in one parameter. For that purpose, the continuation

algorithm is combined with a shooting method adapted to the special demands of high-order systems. For details, see [24].

Parameter Estimation and Optimization

Every model of chemical engineering processes relies on parameters connecting

mathematics with reality. Two different problems arise in this context. The parameter 

estimation problem refers to the determination of unknown model parameters from

measurement data. The improvement of process behavior results in the parameter 

optimization problem. Both of these problems can be solved by Diva. In any case itis crucial to compute parameter sensitivities W i,j  = ∂xi /∂pj , i.e., partial derivatives

of state variables x with respect to parameters p. This information is obtained by

differentiating the model equations (13.1) with respect to the parameters p yielding

the equations of variations

BdW 

dt =

∂

∂x

f  − B

dx

dt 

· W  +

∂

∂p

f  − B

dx

dt 

. (13.4)

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The DAEs (13.1) and (13.4) are solved simultaneously in Diva with the integrator

SDASAC [3].

The parameter estimation problem is formulated as a least-squares (LS) problem

where the squares of the differences between measured and calculated output vari-

ables at discrete points of time are minimized. In Diva, this problem is solved by

the sequential quadratic programming (SQP) algorithm E04UPF from the NAG li-

brary [27] which uses sensitivities calculated according to Equation (13.4). Parameter

estimation for a steady-state model is also possible. In this case, the sensitivities W i,j 

are calculated by finite differences with one of the solvers mentioned above. Nor-

mally, only a subset of the desired set of parameters can be estimated at the same time

with a defined accuracy. This subset is determined in Diva by a parameter analysis

based on the calculation of output sensitivities ∂y/∂p [22]. Combining this parameteranalysis with parameter estimation provides a powerful tool to determine unknown

model parameters from measurement data.

The optimization of processes and plants is an important engineering task in the

design stage as well as in the retrofit of existing plants. Mathematically, this leads to

a parameter optimization problem where an objective function has to be minimized

minp,u

=  t end

t start

g(p,u(t),x(t),t)dt + h(p, u(t end),x(t end), t end) (13.5)

pmin ≤ p ≤ pmax

withparameters p, inputvariables u, and state variables x. Thecriteriong is calculated

over the considered time horizon t  ∈ [t start, t end], whereas h considers only the final

point t end. In the case of a steady-state optimization problem, g equals zero and only

h is to be minimized. For a dynamic optimization problem the input trajectories

u(t) are discretized in order to obtain a parameter optimization problem where only

time-independent values of parameters have to be determined. As an illustrating

example, the dynamic optimization of coupled distillation columns is given in [37].

This class of problems is solved in Diva with the SQP routine E04UCF from the

NAG library. The corresponding model equations and sensitivity equationsare solved

with the integrator SDASAC in the case of dynamic optimization or with one of the

nonlinear equation solvers in the steady-state case. The influence of parameters on

the optimal solution is quite different. To choose the most promising parameters as

optimization variables, a sensitivity analysis can be applied. Like in the parameterestimationproblem, thecombination of analysis andoptimizationprovides a powerful

tool to determine parameters and input trajectories in order to improve the process

performance.

Due to the utilization of sparse matrix techniques and standard numerical routines

like Harwell [9] and BLAS, Diva is suitable for large-scale systems with as much as

10.000 state variables and with over 100.000 Jacobian entries. Systems of this scale

have been realized in Diva with reasonable computation times.

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13.2.2 Code Generation of Diva Simulation Models

For an efficient and user-friendly implementation of process unit models, the code

generator (CG) and a CG language have been developed [15, 31]. The syntax of theCG language is similar to the programming language LISP [36] which is used for the

implementation of the CG. As depicted in Figure 13.1, the CG input file serves as an

interface for the symbolic preprocessing tool, the process modeling tool, as well as for

the user. The CG language provides powerful definition elements for the automatic

generation of compact and efficient Fortran code by the code generator. The two-

step implementation of simulation models by the symbolic model definition with

the CG language and its transformation into Fortran code has several advantages.

On one hand, the implementation of model equations in the CG language is much

more convenient to the user than the direct coding in Fortran. For example, thecoding of the Jacobians’ patterns required by the sparse matrix numerics of Diva is

automatized by the code generator. Doing this manually would be a cumbersome

and time consuming task. On the other hand, the executable Fortran program

includingtheautomatically generatedsimulation models leads to bettercomputational

performance than interpretation of model equations during runtime.

13.2.3 Symbolic Preprocessing Tool

The capabilities of the symbolic preprocessing tool SyPProT allow transforma-

tion of process models derived from first principles into linear implicit DAE models

required by Diva. The functionalities concern MOL discretization of PDE and IPDE

into DAE, index analysis and reduction of DAE, as well as DAE transformation

into the linear implicit form (13.1). SyPProT is implemented with the computer-

algebra-system Mathematica and it is designed as a toolbox that contains several

preprocessing modules. The architecture of the preprocessing tool is depicted in

Figure 13.4. On the left, it shows two files for symbolic preprocessing model repre-

sentation using the Mathematica data structure (MDS) and a CG input file, which

is the interface to the Diva simulation kernel.

MOL-Discretization

Index-Analysis

and -Reduction of DAE

DAE-Transformation into

MATHEMATICA

MDS-

Writer

Reader

MDS-

SyPProT Command PaletteMATHEMATICA

Notebook Front End

DAEPDE, IPDE

MATHEMATICA

MathematicaData Structure File

 MOL-Parameter}

{PDE, IPDE, DAE Model,

MathematicaData Structure File

{DAE-Model}

MATHEMATICA

CG-LANGUAGE

CG Input File

{DAE: B, f, x, u, y, p, t} Symbolic PreProcessing Tool SyPProT

CG-

Writer

FIGURE 13.4

Architecture of the symbolic preprocessing tool SyPProT [14].

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The symbolic preprocessing tool SyPProT is depicted on the right as part of the

computer-algebra-system Mathematica. SyPProT contains three preprocessing

modules for MOL discretization of PDE and IPDE, for index-analysis and -reduction

of DAE,andforDAE transformation into thelinear implicit form (13.1). Furthermore,there are interface modules for the interpretation of MDS objects (MDS-Reader) and

for file output (MDS-Writer, CG-Writer). The MDS model definition is interpreted

by the MDS-Reader. The MDS-Writer generates an MDS file of a selected model

definition stored in the data management. The CG-Writer is used to translate the

preprocessing result of linear implicit DAE models from MDS into CG-language.

The interface and the preprocessing modules are interconnected by the data manage-

ment. It calls the preprocessing modules and handles the model information of all

preprocessing steps.MDS models can be defined as a text file or interactively within theMathematica

notebook front end which is shown at the top within the Mathematica box. The

notebook front end contains as an extension of the Mathematica standard palettes

the SyPProT command palette. This palette allows a convenient generation of MDS

objects as well as the execution of the preprocessing commands. By means of the

notebook front end, complete preprocessing sessions as well as additional symbolic

manipulations using theMathematica capabilities can be performed and saved.

13.2.4 Computer-Aided Process Modeling

The knowledge-based process modeling tool ProMoT supports modelers in devel-

oping Diva models on the phase level of a process unit [41, 40]. The object-oriented

knowledge base of ProMoT contains modeling entities for the representation of the

structure of process unit models, the model equations, and the occurring material

substances and mixtures. These structural, behavioral, and material model entities

are defined with the model definition languageMdl of ProMoT. A process model is

represented asMdl frames in anMdl-file which can be generated using a text editoror the graphicalMdl-editor. ProMoT is implemented in Lisp and its object-oriented

extensionClos. This allows ProMoT direct access to the code generator commands.

With the CG-Writer of ProMoT, a CG input file can be generated. Future research for

ProMoT will include the extension of the language concepts and constructs of Mdl

for an implementation of PDE and the connection of ProMoT and the preprocessing

tool SyPProT.

13.3 MOL Discretization of PDE and IPDE

Typical equations for distributed parameter models of chemical processes are cou-

pled PDE and IPDE with one space or property coordinate z for spatial distributed

models or population models, respectively. The PDE and IPDE are up to second

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order in the space or property coordinate z and of first order in time t . The initial

conditions (IC) are specified by the profiles x0(z).

A ∂x∂t 

= C ∂2x∂z2

+ ∂F ( x , u , p, z, t)∂z

+ S(x, u, p , z, t ) t > t  0, z ∈ (0, l) (13.6)

x(z,t 0) = x0(z) z ∈ [0, l] . (13.7)

The related boundary conditions (BC) depend on the input vectors v0,l (t):

0 = C0,l∂x

∂z

+ F 0,l (x,v0,l , p, t ) t > t  0, z ∈ {0, l} . (13.8)

The matrices A(x,u,p,z,t) and C(x,u,p,z,t) as well as the source vector

S(x,u,p,z,t) may depend on the state variables x(z,t) ∈ Rn, the input variables

u(z,t), the parameters p, and on z and t . The flux function F ( x , u , p, z, t) represents

the convective transport with the flow velocity w(x,u,p,z,t) according to:

F ( x , u , p, z, t) = w(x,u,p,z,t)x(z,t) . (13.9)

In case of population models [30], the source vector S consists of non-integral terms,

collected in the function vector R(x,u,p,z,t) and integral terms concerning the

function vector Q(x,u,p,z,z, t).

S(x,u,p,z,t) = R(x,u,p,z,t) +

zMAX zMIN

Q(x,u,p,z,z, t) dz . (13.10)

The function vector R represents the sources and sinks only depending on the current

value of the population coordinate z. The remote effects of the population alsoconcerning interactions of several particles are described by the integral over the

functionvector Q. The integral limitszMIN and zMAX dependontherelated population

effects and are either the population domain boundary values 0 or l, the coordinate z

or a function of this coordinate f(z) ∈ [0, l].

These general forms of the PDE, IPDE, and BCs which describe hyperbolic as well

as parabolic problems are handled by the symbolic preprocessing tool SyPProT. It

is assumed that the model equations are well posed and have a unique solution.

For the discretization of PDE and IPDE models (13.6) through (13.8), the MOL

approach discretizes the domain of the independent variable z for the space or property

coordinate. The continuous coordinate z is replaced by discrete grid points zk or zk(t),

k = 1(1)kmax for static or moving grids, respectively, which is illustrated for a static

grid in Figure 13.5. The grid partition also includes the segmentation by means of 

continuous control volumes (CV). In this case, the grid points represent the cell center

pointsof the CVs. On the discretizeddomainof z, thepartial derivatives ∂ i x(z, t)/∂zi ,

i = 1, 2 are approximated by functions of the state variables xk(t) at the grid points

zk . For these approximations, SyPProT provides configurable finite-difference and

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MOL

   B   C

ODE

IC IC   O   D   E  o  r   A

   E

   O   D   E  o  r   A

   E

   B   C

IPDE

PDE

FIGURE 13.5

Method-of-Lines (MOL) approach for the transformation of partial differential

equation (PDE), integro partial differential equation (IPDE), and the related

boundary conditions (BC) into ordinary differential and algebraic equations(ODE,AE) [32].

finite-volume schemes. The result of the MOL discretization are ODEs at the inner

grid points zk , k = 2(1)kmax − 1 and algebraic equations (AE) or ordinary differential

equations (ODE) at the boundary grid points zk , k = 1, kmax for finite-difference or

finite-volume schemes, respectively. Furthermore, the ICs x(z, 0) are transformed

into xk(0), k = 1(1)kmax, which are consistent with the AEs if the ICs (13.7) andBCs (13.8) are formulated consistently.

In the next subsections, the discretization schemes available in SyPProT are de-

scribed. First the standard MOL discretization schemes of finite differences (FD)

and finite volumes (FV) are presented. However, these FD and FV schemes have

some disadvantages when applied to problems with steep moving fronts. High-order

approximations lead to unphysical oscillatory behavior of the solution. Low-order

schemes do not show such oscillations but require a very fine grid for sufficiently

accurate solutions. Due to the high number of grid points, the computational effort

is quite large.

There are two approaches to combine the objectives of high numerical accuracy

as well as efficient computation: so-called high-resolution schemes [8] and moving-

grid methods [5, 6]. High-resolution schemes use high-order approximations based

on a suitable choice or weighting of the approximation points to get an oscillation-

free scheme. The adaptivity concerns the approximation polynomials. Examples are

essentially-non-oscillatory (ENO) and flux-limiter schemes [33, 34, 38, 16] which

are also described in a following subsection.

Moving grid methods use variable grid nodes to concentrate these nodes in solution

sections with steep fronts. The grid nodes are placed where they are most needed

to keep the discretization error small. The adaptivity of this approach concerns the

flexible distribution of the available number of grid nodes. Furthermore, moving grid

methods are distinguished into dynamic and static regridding methods. Dynamic

regridding methods operate on continuously moving grid nodes. The number of 

state variables and equations is increased by the number of moving grid points. Static

regridding methods compute the solution for a certain number of time steps on a static

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grid. Then the regridding step calculates new positions of grid points, whereas the

insertion and elimination of grid points is possible. After each such static regridding

step, an interpolation is necessary to transform the solution on the old grid to a solution

on the new grid. The interpolation introduces additional numerical errors. Moreover,in order to continue the simulation, consistent initial values must be computed. Of 

course both regridding techniques can be combined.

In the context of automatic discretization with the symbolic preprocessing tool of 

the simulation environment Diva, the application of static regridding methods seems

to be less suitable considering the static storage management of the Fortran-based

Diva simulation kernel, which prohibits a change in the number of state variables

and equations during simulation. A further disadvantage is the computational cost

of frequent initialization of the model equations required after each static regriddingstep, especially for multistep integration procedures. For the symbolic preprocessing

tool SyPProT, we focus on a dynamic regridding method based on the Lagrangian

approach [43] which is comparably easy and flexibly implemented within the software

architecture of SyPProT. This method is described in the last part of this section.

13.3.1 Finite-Difference Schemes

The finite-difference (FD) schemes used in the symbolic preprocessing tool are

based on Lagrange polynomials and can be applied on uniform or non-uniform gridsfor the discretized space coordinate. The spatial grid is defined by a grid function

z(k) that leads to the grid points zk , k = 1(1)kmax. Then the state variables x(z,t)

are approximated by Lagrange polynomials according to

x(z,t) ≈ L[r,s](z,t) =

sj =r

lj (z)xj (t) . (13.11)

This approximation uses the state variables xj (t), j  = r(1)s at the grid pointsbetween zr  and zs  (Figure 13.6). The polynomial (13.11) is used to approximately

calculate the spatial derivatives of x(z,t) at the center point zk by differentiating with

respect to z:

∂i x(z,t)

∂zi

z=zk

≈∂i L[ri ,si ](z,t)

∂zi

z=zk

. (13.12)

For the approximation of an integral occurring in IPDE problems (13.10), the integral

range is split into subranges for each grid point i = iMIN(1)iMAX with z−i = zMIN

for i = iMIN and z+i = zMAX for i = iMAX. For the subrange integrals, the state

variables are approximated by Lagrange polynomials according to (13.11):

 zMAX

zMIN

Q(x,u,p,z,z,t)dz =

iMAXi=iMIN

 z+i

z−i

Q(L[ri ,si ](z, t ) , u , p , z, z, t) dz .

(13.13)

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FIGURE 13.6

Lagrange polynomial (13.11) for the approximation of x(z,t) in the range zr <

z < zs .

This leads to the discretization parameters ri , si , and k for the order of the spatial

derivative i = 0, 1, 2. For the symbolic preprocessing, these discretization parame-

ters can be individually defined for each PDE and IPDE. The approximations at the

boundary points require a special treatment. At the boundary points, the approxi-

mation polynomials can depend on grid points outside the spatial domain. This is

avoided by means of the so-called sliding differences technique. Thereby local spatial

oscillations possibly appearing in numerical solutions of hyperbolic problems can be

prevented by an order reduction of the approximation polynomials.

13.3.2 Finite-Volume Schemes

The finite-volume (FV) method subdivides the spatial domain into a finite number

of discrete contiguous control volumes (CV) to which the PDE or IPDE (13.6) are

applied. This leads to the integral form of the PDE or IPDE on which the FV method

is based [29].

For the CV definition, a uniform or non-uniform grid function z(k), k = 1(1)kmax

is used. The grid function can either define the cell center points of the CVs or the cell

boundary points. This leads to the two possible grid practices shown in Figure 13.7.

For the grid practices named “GridPoints” and “VolumeBounds,” the grid functions

compute the cell center points and the cell boundaries, respectively.

FIGURE 13.7Definition of the control volumes (CV) for discretization by finite volumes and

of the corresponding grid practices “GridPoints” (left) and “VolumeBounds”

(right).

The FV discretization of a PDE is performed in two steps: the integration over each

CV and the approximation of the resulting cell boundary values. In order to illustrate

this procedure, a scalar convection diffusion PDE for x(z,t), z ∈ (0, l) is used with

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the source term S(x,u,p,z,t).

∂x

∂t  = −v

∂x

∂z + D

∂2x

∂z2 + S t > 0, z ∈ (0, l) . (13.14)

The related BC and IC are not considered. The integration of PDE (13.14) over the

kth CV from the cell boundaries z−k to z+

k (for notations see Figure 13.8) leads to

z+

k − z−k

dxk

dt = −v

x+

k − x−k

+ D

∂x

∂z

z+k

−∂x

∂z

z−k

+

z+k − z−

k

S k

(13.15)

with

S k = S (xk, uk, p , zk, t) = Rk +

iMAXi=iMIN

 z+i

z−i

Q

x , u , p , z, z , t  

dz (13.16)

= Rk +

iMAX

i=iMIN z+

i − z−i

Q (xk, uk, p , zi , zk , t ) .

The discretized source S k is interpreted as a representative mean value for the CV k.

FIGURE 13.8

Profile assumptions of  piecewise constant (upwind) (left) and piecewise linear

schemes (right) for the approximation of x±k and the derivatives ∂x/∂z|z±

k, re-

spectively.

The integral term included in (13.17) is discretized by a sum of subintegrals. These

subintegrals represent the involved CVs according to the complete integral rangein (13.10) with z−

i = zMIN for i = iMIN and z+i = zMAX for i = iMAX.

In the integral formofPDE (13.15), the values of x±k and ∂x/∂z|z±

kareunknownand

must be approximated. For the boundary CVs, the boundary conditions are inserted

into (13.15) to eliminate unknown values at the inlet and the outlet of the spatial

domain. For the approximation of the remaining unknown values at the interior cell

boundaries z−k and z+

k , so-called profile assumptions are used as shown in Figure 13.8.

The following ODE for the kth CV is obtained applying these profile assumptions

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to (13.15).

dxk

dt 

zk = −v (xk − xk−1) + Dxk+1 − xk

δzk

−xk − xk−1

δzk−1 + S kzk . (13.17)

Here, the symbols zk , δzk , and δzk−1 are the distances of the CV boundary points

and the CV center points, respectively.

For IPDE of population balances, the discretization of an integral within the source

term S  introduces further difficulties which concern the moment conservation of 

population distributions. More details about this topic as well as a solution strategy

are described in [18].

13.3.3 High-Resolution Schemes

The concept of high-resolution schemes is based on approximation polynomials

of high order with variable selection or weighting of the approximation points. This

adaptive idea allows combination of high numerical accuracy and stability. High-

resolution schemes are derived for hyperbolic conservation laws, which can be written

in the following form using the PDE notations (13.6):

A∂x

∂t  =∂F ( x , u , p, z, t)

∂z + S(x, u, p, z, t) t > 0, z ∈ (0, l) . (13.18)

High-resolution schemes are mostly used in the context of finite-volume discretiza-

tion which integrates (13.18) over the CV k according to

 z+k

z−k

Adx

dt dz =

F +k − F −k

+

 z+k

z−k

S dz (13.19)

with the flux functions F ±k = F ( x , u , p, z, t)|z=z±k at the cell faces of the CV, see

Figure 13.8. The focus is directed on the approximation of these flux function values.

This approximation can also be interpreted as a more sophisticated profile assumption

for the flux function.

Two high-resolution schemes are briefly described: the ENO-Roe scheme [33, 34]

and the so-called robust upwind scheme [16].

13.3.3.1 ENO-Roe Scheme

The ENO-Roe scheme [33, 34] uses the so-called ENO interpolation which ap-

proximates the flux values using a fixed number of interpolation points on solution-

dependent positions. The adaptive aspect of this scheme is the detection of the “op-

timal” position of the interpolation points. This position is determined successively

by the construction of a Newton interpolation formula of increasing order for approx-

imation of the flux. This approximation procedure compares divided differences of 

the Newton interpolation formula built of adjacent interpolation points. The minimal

magnitude of the compared divided differences determines the position of the next

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interpolation point. ENO schemes can be extended to any approximation order. But

for schemes higher than third order, numerical stability problems increase [44]. An

important advantage of the adaptive determination of the approximation points is that

a reversal of the flux direction is automatically considered.

13.3.3.2 Robust Upwind Scheme

The robust upwind scheme [16] is a κ-interpolation scheme originated by Van

Leer [19] with a modified flux limiter of Sweby [38]. Its accuracy is of 2nd to 3rd

order. The robust upwind approximation of F +k uses flux function values F j  =

F (xj , uj , p , zj , t) of the upwind located CV centers j  = k, k − 1.

F +

k = F k + 0.5φ r+k (F k − F k−1) , (13.20)

r+k =

F k+1 − F k +

F k − F k−1 + , (13.21)

φ(r) = max

0, min

2r, min

1

3+

2

3r, 2

. (13.22)

The limiter function φ depends on the ratio of consecutive solution gradients r +k .

The parameter is very small to avoid division by zero in uniform flow regions.

In contrast to the ENO scheme, the approximation points zk are fixed. The adap-tation lies in the solution dependent weighting of the approximation points by the

limiter function. The described high-resolution schemes can be easily integrated into

the MOL module of SyPProT, as shown in Section 13.4.2.

13.3.4 Equidistribution Principle Based Moving Grid Method

The considered moving grid method within the symbolic preprocessing tool uses

the Lagrangian approach of grid nodes, which are continuously moved along with

the solution. The grid consists of moving grid nodes zk(t), k = 2(1)kmax − 1 for the

inner spatial domain and fix grid nodes zk(t), k = 1, kmax at the domain boundaries.

This approach reduces the rapid changes of steep front solutions at fixed grid points

leading to small time steps during numerical simulation. Thus, the additional effort

of the moving grid discretization is partly compensated.

Using the Lagrangian form of the time derivative at the grid node zk(t)

∂x

∂t  zk (t)

=dxk

dt 

−∂x

∂z

dz

dt zk (t)

, (13.23)

the semi-discrete form of Equation (13.6) at the grid node zk(t) can be written as

A

dxk

dt −

∂x

∂z

dz

dt 

zk (t)

= (13.24)

C∂2x

∂z2

zk (t)

+∂F ( x , u , p, z, t)

∂z

zk (t)

+ S(xk, uk, p , zk, t) t > 0 .

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According to this, no transformation is necessary for the BC (13.8). Additional equa-

tions are required for the computation of moving grid nodes zk(t), k = 2(1)kmax − 1.

The general form of the moving grid equations reads

τ Edz

dt = G(x,z,κ) (13.25)

with the temporal regularization parameter τ , the matrix E(x,z,κ), and the function

vector G(x,z,κ). The parameter κ is used to control the grid expansion such that

κ

κ + 1≤

zi − zi−1

zi+1 − zi

≤κ + 1

κ. (13.26)

The matrix E as well as the function vector G are based on the spatial equidistri-bution equations zk+1

zk

M(z,x)dz =

 zk

zk−1

M(z, x)dz k = 2(1)kmax − 1 (13.27)

with the arc-length monitor function M(z,x) considering the spatial gradients of the

state variables x ∈ Rn

M(z,x) =1 + 1

n

ni=1

∂xi

∂z

2

. (13.28)

The elements of E and G depend on temporal as well as spatial smoothing tech-

niques. For the considered implementation within the symbolic preprocessing tool,

the smoothing procedures following Dorfi and Drury [5] are taken. A more detailed

discussion of this smoothing technique can be found in [43].

The discretization of Equations (13.27) and (13.28) depends on the chosen dis-

cretization of the spatial domain according to finite differences or finite volumes aswell as on stability criteria [21].

13.4 Symbolic Preprocessing for MOL Discretization

The preprocessing procedure of  SyPProT for automatic MOL discretization of 

distributed parameter models is performed in three steps. The first step is the MOL

discretization of PDE and IPDE into DAE by means of a correspondingMathema-

tica module of the preprocessing tool, see Figure 13.4. The available discretization

methods of finite differences and finite volumes as well as the equidistribution prin-

ciple based moving grid method have been described in the previous section. In a

second step, the resulting DAE are transformed by another preprocessing module

into the linear implicit model form (13.1) required by Diva and the CG. Finally, the

CG-Writer generates the CG input file (Figure 13.4).

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For application of the automatic MOL discretization by the symbolic preprocessing

tool, the model as well as required MOL parameters must be defined. The Mathe-

matica data structure (MDS) is a tailor-made input format for this purpose. This

section describes the definition of PDE and IPDE models as well as that of MOLparameters by means of the MDS. Furthermore, the preprocessing module for MOL

discretization is explained. Therefore, the internal procedure for application of the

implemented discretization schemes is presented.

13.4.1 Mathematica Data Structure

For the definition of mixed PDE, IPDE, and DAE models and the related MOL pa-

rameters, the symbolic preprocessing requires appropriateMathematica data struc-ture (MDS) objects containing the complete model information (Figure 13.4). The

MDS is based on the computer-algebra-systemMathematica and uses (asMathe-

matica itself) only a small number of symbolic programming methods. Mainly the

methods of expressions, rules, and lists build the framework for the MDS. Further-

more, the MDS and the according MDS input file are based on the CG language and

the CG input file, respectively, to facilitate the final transformation of a preprocessed

PDE and IPDE model into a CG input file. The MDS provides several functions for

the definition of mixed PDE, IPDE, and DAE models as well as MOL parameters.

Each MDS definition function builds a section within the MDS input file, which con-

tains the whole information about the model and the related MOL parameters. The

MDS definition functions can also be executed directly in the notebook front end

of Mathematica forcing the MDS-Reader to generate the according MDS objects

(Figure 13.4).

13.4.1.1 Definition of Mixed PDE, IPDE, and DAE Models

The definition functions for a PDE and IPDE model comprise a general modeldescription and the definition of parameters, variables, and equations. The equation

section SystemEquations[...] contains PDE, IPDE, and DAE in separate

subsections but in a standardized symbolic representation. Special MDS functions

are provided for the coupling of sequential and parallel neighboring spatial domains.

Sequential domaincoupling requiresproper inner boundaryconditions with consistent

discretization schemes according to the PDE in the involved spatial domains. In order

to obtain a common discretization scheme for a PDE or IPDE as well as the according

boundary and inner boundary conditions, the definitions of the PDE or IPDE, the

boundary, and inner boundary conditions are grouped together into a common section.In addition, this common section specifies common MOL parameters.

Coupling of parallel spatial domains requires interpolations between different grid

point positions, if different individual spatial grids are used for the involved domains.

The MDS provides so-called shifted grids for parallel coupled domains. A shifted

grid is identical to the related basic grid with the only difference being an offset in the

values of the grid points. Thereby, interpolations are not necessary. An additional

advantage of a shifted grid is that it does not require additional own state variables and

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corresponding new equations if a moving grid method is used. Due to the identical

grid node distribution of the basic and the shifted grid, only the basic grid extends the

number of state variables and equations. This reduces the complexity increase of the

moving grid approach. The use of a shifted grid is demonstrated in Section 13.5.1for the example of a circulation-loop-reactor model.

13.4.1.2 Definition of MOL Parameter

For the MOL parameters, two definition functions exist: the function

Domains[...] defines the grid functions of the spatial domains and the function

Discretizations[...] defines the discretization methods and its parameters.

The MDS definition function Domains[...] for a spatial grid contains argu-

ments for the grid name, and the grid function as well as its parameterization. Thegrid function is aMathematica expression computing the grid node location for any

valid index value. The grid function must be reversable to compute the index value

for a given location within the spatial domain. This is required for graphical output

or process unit coupling. The moving grid method is activated by the MovingGrid

attribute. Its value is a list of moving grid parameters. All moving grid parameters are

assigned default values which are used if the user defines an empty list for the moving

grid parameters. But the user can also redefine these values to tune the moving grid

method. The following moving grid parameters exist, see Section 13.3.4:• SpaceSmoothing defines the spatial smoothing parameter κ (default value:

κ = 2),

• TimeSmoothingdefines the temporal smoothing parameter τ  (default value:

τ  = 0),

• MonitorStates collects the state variables of this domain to be considered

for the monitor function; the default configuration considersall distributed state

variables of the considered domain.If the moving grid method is activated, the grid function computes only the initial

distribution of the grid nodes.

The available discretization methods for the Discretizations[...] func-

tion are configurable FD and FV schemes, which are defined by the MDS func-

tions FDMethod[...] and FVMethod[...], respectively. The FD schemes

define individual Lagrange polynomials for approximation of the spatial derivatives

∂ i x(z, t)/∂zi , i = 0, 1, 2. For the configuration of the Lagrange polynomials, the

MDS provides three MOL parameters to determine the discretization parametersri , si , k, i = 0, 1, 2 (see Section 13.3.1):

• PolynomPoints mi defines the numbers of interpolation points mi =

si − ri + 1,

• Eccentricityqi determines theapproximation centers k = (ri +si )/2+qi ,

• OrderReduction oi defines the order reduction of the approximation poly-

nomials in the boundary area.

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The FV scheme is defined by the MDS function FVMethod[...], which is

configured by the choice of profile assumptions for approximations of the unknown

CV boundary values (see Section 13.3.2). The MDS provides thereto the parameter

Profile.Extracts of the MDS functions SystemEquations[...], Domains[...],

and Discretizations[...] of the MDS input files for the circulation-loop-

reactor model and the true-moving-bed process unit model are presented in Sec-

tions 13.5.1 and 13.5.2.

13.4.2 Procedure of the MOL Discretization

For a Diva simulation of a PDE or IPDE model, three preprocessing steps haveto be performed (see SyPProT box in Figure 13.4 f or corresponding preprocessing

and interface modules): the MOL discretization of PDE and IPDE into DAE (top

right), the transformation of the resulting DAE model into linear implicit form (13.1)

(bottom right), and the generation of a CG input file of this DAE model (bottom left).

The user can execute these steps by two commands within aMathematica session.

The command AutomaticDiscretization[...] performs the loading of an

MDS input file and the execution of the discretization procedure. The result is a DAE

model in MDS. This DAE model is transformed into form (13.1) and then writteninto a CG input file by the command MDS2CG[...].

The internal procedure of the MOL discretization in the symbolic preprocessing

tool is described in  Figure 13.9. The gray boxes with solid lines mark different

Mathematica modules. The module for discretization management collects the

information required for the discretization of a PDE or IPDE and the related BC as

defined in the MDS input file. In particular, information on the grid function of the

according spatial domain and the selected discretization method is supplied.

From the symbolic manipulation point of view, the moving grid method describedin Section 13.3.4 can be interpreted as a preliminary model transformation before

the actual discretization. This allows a comparably simple integration of this method

in the discretization procedure of the preprocessing tool. The moving grid module

applies Equation (13.23) to transform PDE and IPDE (13.6) into a form according

to Equation (13.24). For the computation of the moving grid nodes, the additional

moving grid equations (13.25) are required. These equations are automatically gen-

erated considering the grid expansion constraint (13.26), the spatial equidistribution

principle (13.27) based on the arclength monitor (13.28), and temporal grid smooth-

ing techniques. Then the transformed equations are discretized on the moving gridby FD or FV schemes, which are implemented in separate modules.

The FD and FV modules generate specific auxiliary variables for the generation

of shorter and more efficient simulation models. Such auxiliary variables are used to

compute, e.g., distances of grid nodes or lengths of CVs. The FD module approx-

imates the state variable functions and their spatial derivatives by Lagrange poly-

nomials (13.11) and their spatial derivatives (13.12), respectively. The FV module

integrates the PDE or IPDE over each CV under consideration of the boundary con-

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grid smoothing procedures

based on spatial equidistribution principlemoving grid equations

grid definition and discretization method selection

discretization management 

PDE / IPDE + BC

moving grid module 

integration over allapproximation of

with

Lagrange-polynomials

approximation ofapproximation of controlvolume boundary values

 x(z,t)

generation of method specific auxiliary variables

control volumes under

consideration of the BC

DAE

finite-differences module finite-volumes module 

piecewiseconst./linear

schemes

high

schemesresolution

equation transformation on moving grid coordinate

FIGURE 13.9

Procedure for use of the MOL discretization of partial differential equations

(PDE), integro partial differential equations (IPDE), and boundary conditions

(BC) in SyPProT.

ditions and approximates the resulting unknown values at the CV boundaries by

piecewise constant or piecewise linear profile assumptions (Figure 13.8). The high-

resolution schemes described in Section 13.3.3 could also be used to approximate

the CV boundary values of the flux functions, see Section 13.3.3. These schemes

are represented by the gray box with dashed lines in Figure 13.9, as they are not yet

implemented in the finite-volume discretization module.

13.5 Application Examples

For illustration of the MOL capabilities of the symbolic preprocessing tool and the

numerical methods inDiva, two application examples from chemical engineering are

presented. The first example is a circulation-loop reactor. The reactor consists of se-

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quential and parallel coupled spatial domains. The model is described by PDE, which

are coupled by inner boundary conditions for sequential neighboring domains and by

the heat exchange between the parallel adjacent domains. The MOL discretization

is performed with FD schemes applied on a moving grid. The second example is amoving-bed chromatographic process. The PDE describes the counter-current flow

of two phases. FV schemes are used for automatic MOL discretization.

13.5.1 Circulation-Loop-Reactor Model

The circulation-loop reactor (CLR) [13, 23] represents a complex example for the

application of the MOL. A scheme of the reactor is shown in Figure 13.10. A special

feature of the reactor is the geometrical construction. The reactor consists of threespatial domains. Each domain is described by one dynamic PDE for the energy bal-

ance and two PDE for the material balances based on a quasisteady-state assumption.

Besides the boundary conditions at the inlet and the outlet of the reactor, there are

inner boundary conditions for the coupling of sequential neighboring domains. The

dynamic behavior of the reactor shows moving fronts of different steepness, which re-

quire discretization schemes of adequate accuracy. In the following, the CLR model

equations, extracts of the MDS input file for symbolic preprocessing of the PDE

model, and the simulation results of Diva are presented.

catalytic fixed bed

zheating outer tube

inner tube

loop

FIGURE 13.10

Circulation-loop reactor consisting of an inner tube, the reactor loop, and an

outer tube; the inner and outer tube are built as a co-current heat exchanger

[13, 23].

13.5.1.1 Model Equations

The CLR consists of three spatial domains comprising the inner tube, the loop, andthe outer tube. A common coordinate z describes the 1-dimensional spatial domains:

Inner tube (I ) : z ∈ [0, ltube]

Reactor loop (L) : z ∈ [ltube, lloop + ltube]

Outer tube (O) : z ∈ [ltube, lloop + 2ltube] .

(13.29)

The reactor is modeledas a single-process unit. Its parts are structured as sequential

and parallel adjacent spatial domains. The reactor modules are shown in Figure 13.11.

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The loop domain is coupled with the tube sections by inner boundary conditions at

z = ltube and z = lloop + ltube, respectively. The inner and outer tube domains are

parallel adjacent domains and coupled by the heat flux over their interface. For the

outer tube domain, a so-called shifted grid is used (see Section 13.4.1). This grid isshifted and uses the grid of the inner tube domain as a basic grid but with an offset of 

z = lloop + ltube.

input

output

IBC

IBC

FIGURE 13.11

Scheme of the circulation-loop reactor with three spatial domains: inner tube

(I ), reactor loop (L), and outer tube (O). The domains are coupled by inner

boundary conditions (IBC) and the heat exchange ().

The model equations of the CLR are formulated for every spatial domain j  ∈

{I , L , O} according to (13.29). The PDE for the temperatures T j (z,t) and the com-

ponent mole fractions xij (z,t) are derived from energy and material balances as:

ρcp

s

∂T j 

∂t = −

ρcp

g

vj  ∂T j 

∂z

+ λ

∂2T j 

∂z2 + Qj 

ex +

2i=1

−hR,i ri xj 

i , T j  (13.30)

0 = −vj  ∂xj i

∂z−

M g

ρgri

x

j i , T j 

i = 1, 2 , j  ∈ {I , L , O} . (13.31)

The expressions for the heat exchange rates Qj ex (z,t), the velocities vj , and the

reaction ratesri (xj i , T j ) canbefound in [23]. The corresponding boundaryconditions

at the inlet at z = 0 are given byρcp

g

vI 

T in − T I 

z=0

+ λ

∂T I 

∂z

z=0

= 0 , xi,in − xI i

z=0

= 0 (13.32)

and at the outlet of the reactor at z = lloop + 2ltube by

∂T O

∂z

z=lloop+2ltube

= 0 . (13.33)

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Theinner boundary conditions at thetransitionsof thesequential neighboring domains

ensure thermal and material equilibrium and the equality of the fluxes over the inner

boundaries of the reactor model. Here, only the inner boundary conditions at z = ltube

are presented:

ρcp

g

vI  T I 

z=ltube

− λ∂T I 

∂z

z=ltube

=

ρcp

g

vL T L

z=ltube

− λ∂T L

∂z

z=ltube

(13.34)

T I 

z=ltube

= T L

z=ltube

, xI i

z=ltube

= xLi

z=ltube

. (13.35)

The model is completed by appropriate initial conditions T j (0, z) and xj i (0, z).

13.5.1.2 Definition of Model Equations for Symbolic Preprocessing

For thedefinitionof model equationsforsymbolicpreprocessing, theMDSfunction

SystemEquations[...] is used. The following extract shows the definition of 

the energy balance (13.30), the related boundary condition (13.32), and part of inner

boundary condition (13.34).

1 SystemEquations[

2 Distributed[

3 Domain -> "Tube[z]",

4 Comment -> "reaction zone of the inner and outer tubes"

5 Scope -> { ...

6 Scalar[ rhocps D[Ti[z,t],t] == - rhocpg vi D[Ti[z,t],z]

7 + lam D[Ti[z,t],{z,2}] + Qpex[z,t]

8 - Summands[dhr[j] rreak[j][z,t],{j,1,NC}],

9 LowerBound ->

10 0 == rhocpg vi (Tin[t] - Ti[z,t]) + lam D[Ti[z,t],z],

11 UpperBound ->

12 Flux[TiOut[t]] == rhocpg vi Ti[z,t] - lam D[Ti[z,t],z],

13 Name -> "PDEInnerTube",

14 Comment -> "energy balance inner tube",

15 Discretization -> "FDOrder24"]

16 ... } ] ...];

Within the MDS expression Distributed[...] in lines 2–16, the equations

related to a spatial distributed domain are defined. The domain itself is specified by

the attribute Domain in line 3. The value of Domain is the name of a user defined

spatial grid, here "Tube[z]". The grid "Tube[z]" defines the grid points of 

the inner tube and also serves as the basic grid for the shifted grid of the outer tube.

Therefore, one can simply add the equations of the outer tube to the attribute Scope

in lines 5–16 of this Distributed object. This is indicated by the dots in line 16.

The Scalar[...] expression in lines 6–15 represents a single PDE with attributes

for relationships at the boundaries (Lowerbound, UpperBound) of the considered

domain and the name of a MOL parameter definition (Discretization) in lines

9, 11, and 15, respectively. The first argument of a Scalar[...] expression is

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the equation itself. In particular, in lines 6–8 the energy balance (13.30) for the inner

tube is shown. Therein, the Mathematica operator D[...] is used for temporal

and spatial derivatives, e.g., D[Ti[z,t],{z,2}] which is the 2nd order spatial

derivative of the state variable Ti[z,t], that represents the temperature T I (z,t).

The attribute UpperBound defines the left part of the inner boundary condi-

tion (13.34). The MDS expression Flux[...] is used to define fluxes at the

boundary of sequential neighboring domains. The object Flux[TiOut[t]] in

line 12 defines the flux at the outlet of the inner tube which is identical with the

flux at the inlet of the loop domain, the right part of Equation (13.34). For the in-

let flux of the loop, another Flux[...] object is defined, which will be equated

to Flux[TiOut[t]] in a separate MDS expression for coupling of sequential

neighboring domains. The value of the attribute Discretization refers to MOLparameters named "FDOrder24". These MOL parameters are used to discretize

the PDE defined above as well as the relations defined in the attributesLowerBound

and UpperBound, which are grouped together in a Scalar[...] expression.

13.5.1.3 Definition of MOL Parameters for Symbolic Preprocessing

The MOL parameters for symbolic preprocessing are defined by two MDS func-

tions. The function Domains[...] defines the grid function of a spatial domain.

The following extract of the MDS input file for the CLR model shows the definitionof the grid for the inner tube domain of the reactor:

1 Domains[

2 Grid[ Tube[z],

3 Computation -> Function[{k},(k-1)*LTube/(kmaxTube-1)],

4 Granularity -> "kmaxTube",

5 MovingGrid -> { SpaceSmoothing -> 4.0,

6 TimeSmoothing -> 0.0,

7 MonitorStates -> {Ti[z,t],To[z,t]} }

8 ], ... ];

The spatial domain is defined by the function Grid[...]. The first argument

of Grid[...] is its name, here Tube[z]. The grid function is specified within

the Computation attribute in line 3. The number of grid points or CV, respec-

tively, is defined by the attribute Granularity in line 4. The moving grid is

activated by the MovingGrid attribute, which defines the moving grid param-eters SpaceSmoothing, TimeSmoothing, and MonitorStates, see Sec-

tion 13.3.4. The monitor function for this moving grid considers only the state vari-

ables Ti[z,t] and To[z,t] representing the temperatures in the inner and outer

tubes.

The discretization method and its parameters are defined in MDS within the ex-

pression Discretizations[...]. The following extract of the CLR example

shows the definition of an FD method by the MDS expression FDMethod[...].

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1 Discretizations[

2 FDMethod[ FDOrder24,

3 PolynomPoints -> {1,3,5}, (* m *)

4 Eccentricity -> {0,1,0}, (* q *)5 OrderReduction -> {0,1,1}] (* o *)

6 ];

In line 2, the FD method is named FDOrder24. This name was already used in the

SystemEquations function presented above. The FD method is specified by the

attribute values of PolynomPoints, Eccentricity, and OrderReduction,

see Section 13.4.1. Each attribute value is a list comprising three values for the

spatial derivative of 0th, 1st, and 2nd order. Here the Lagrange polynomials for

approximation of the spatial derivative of 1st and 2nd order are an upwind-biasedscheme of 2nd order and a central-difference scheme of 4th order, respectively.

13.5.1.4 Simulation Results

The CLR model (13.30) through (13.35) is discretized with the FD discretization

method described in the previous subsection. In particular, a moving grid with 40 grid

points for the tube domain and 50 grid points for the loop domain are used for spatial

discretization. The PDE model comprises 9 PDE, 3 BC, and 6 inner BC. These PDE

are transformed to 390 DAE by the symbolic preprocessing tool. The moving gridmethod adds 90 ODE or AE for τ  = 0 or τ  = 0, respectively. The preprocessed CLR

model is transformed by the code generator to a Diva simulation model.

The simulation results in Figure 13.12 obtained by Diva show the temperature

in the top and the mole fraction x2 in the middle diagrams vs. the space coordinate

z at different time steps. The solutions depict the autonomous periodic operation

characterized by traveling waves with fronts of different steepness [23]. In the left

diagrams, the second reaction front moves from the inner tube domain toward the

end of the loop domain. In the right diagrams, the front moves back against the flowdirection of the fluid.

The grid movement is also shown in Figure 13.12. It clearly shows the grid points

following synchronously the temperature front as well as the fixed grid points of the

domain transitions at z = 0.45m and z = 1.034m. The horizontal lines mark the

times of the spatial solution profiles in the temperature and mole fraction diagrams

above. The simulation has been performed with moving grid parameters κ = 4.0 and

τ  = 0.

A more detailed analysis of the nonlinear behavior of the CLR was performed using

the continuation methods presented in Section 13.2.1.3 [13, 23, 24]. Furthermore,the Matlab interface was used for observer and controller design for non-autothermal

operation [12].

13.5.2 Moving-Bed Chromatographic Process

The second chemical engineering example for MOL application is a moving-bed

chromatographic separation process (MBC). For the continuous isolation of the pure

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FIGURE 13.12

Simulation spatial profiles of temperature and mole fractions over the three do-

mains of the circulation-loop reactor (CLR) as well as the grid-point movement.

components, the process operates with counter-current flow of a liquid and a solid

adsorbent phase. For stable MOL discretization, the reverse direction of the convec-

tive transport in both phases has to be taken into account. This is demonstrated by

the application of FV schemes with upwind and downwind profile assumptions.

With reference to Figure 13.13, the MBC unit consists of four sections bordered

by two inlet nodes (feed and solvent) and two outlet nodes (raffinate and extract). Forsuitable operating conditions, the stronger adsorbed components can be withdrawn

in the extract, the remaining weaker adsorbed components appear in the raffinate.

ThisMBC process isdefined asa flowsheetof several process units onthe plant level

of the Diva simulation kernel. The MOL discretization concerns only one generic

MBC zone which is used four times within the flowsheet. In contrast to the previous

CLR example, the coupling of the adjacent domains is defined by interconnection of 

process units (13.2) instead of inner boundary conditions.

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zone

FIGURE 13.13Scheme of a moving-bed (MBC) process (left) and of a single MBC zone (right)

with the bulk (B), pore (P ), and adsorbent (A) phases as well as the pseudo-

homogeneous phase (K) which contains A and P . The input (cK,i,in, cB,i,in) and

output ( cK,i

zl

, cB,i

zr

) concentrations are also shown in the right diagram.

13.5.2.1 Model Equations

The MBC zones are modeled by a 1-dimensional spatial domain consisting of ahomogenous bed of spherical porous particles (radius Rp) with a constant bed poros-

ity ψ . The mass balances for the components i = 1(1)nc − 1 lead to the following

set of PDE for the concentrations cj,i (z,t), j  ∈ {B, K}. The index B represents

the bulk phase (Figure 13.13). The adsorbent and the pore fluid are combined to a

pseudo-homogeneous phase with index K in Figure 13.13. No mass balances are

required for the solvent concentrations cj,nc (z,t), j  ∈ {B, K}. A detailed descrip-

tion of the modeling and the model assumptions of the considered chromatographic

separation can be found in [35]. The PDE and BC are for the bulk concentrations cB,i ,i = 1(1)nc − 1

∂cB,i

∂t = −v

∂cB,i

∂z+ E

∂2cB,i

∂z2

−3(1 − ψ )

RP ψβ∗(cB,i − cP ,i )

  j f,i

z ∈ (zl , zr ) , (13.36)

0 = E∂cB,i

∂z

zl

− v( cB,i zl − cB,i,in), 0 =∂cB,i

∂z

zr

(13.37)

and for the concentrations cK,i , i = 1(1)nc − 1

∂cK,i

∂t = u

∂cK,i

∂z+

3

Rp(P  + A)· β∗

cB,i − cP ,i

  j f,i

, z ∈ [zl , zr ) (13.38)

cK,i zr= cK,i,in . (13.39)

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Again, the model is completed by appropriate initial conditions. The mass transfer

through the liquid phase from the bulk to the pore (Figure 13.13) is denoted by

j f,i . Parameter β∗ marks the mass transfer coefficient and E is the axial dispersion

coefficient. The symbol v is the interstitial velocity of bulk-fluid and u is the linearvelocity of the solid phase in the counter-current system. Since adsorbent flow is

assumed to be purely convective, only an inlet boundary condition (13.39) exists for

each zone.

The concentrations cK,i (z,t) are defined as

cK,i = εP cP ,i + (1 − εP ) cA,i , i = 1(1)nc − 1 . (13.40)

The concentrations cA,i (z,t) and cP ,i (z,t) are related by the equilibrium isotherm

(αi,j  is the separation factor of two components)

cA,i =αi,nc cP ,i

1 +nc −1j =1

αj,nc − 1

cP ,j 

, i = 1(1)nc − 1 . (13.41)

The complete model for one MBC zone with state variables cj,i (z,t), j  ∈ {B , K , P  },

i = 1(1)nc − 1 comprises 2(nc − 1) PDE, 3(nc − 1) BC, and nc − 1 algebraic

equations.

13.5.2.2 Definition of Model Equations for Symbolic Preprocessing

The following MDS extract shows the material balances (13.36) and (13.38) within

the SystemEquations function.

1 SystemEquations[

2 Distributed[

3 Domain -> "Grid1[z]",

4 Comment-> "the counter-current liquid and solid phases",

5 Scope -> {...

6 Tensor[

7 {Scalar[ D[cB[i][z,t],t] == -v D[cB[i][z,t],z]

8 + E D[cB[i][z,t],{z,2}] +

9 3 (1-psi) beta (cB[i][z,t] - cP[i][z,t])/(Rp psi),

10 LowerBound -> v cBin[i][t] ==

11 v cB[i][z,t] - E D[cB[i][z,t],z],

12 UpperBound -> 0 == D[cB[i][z,t],z],

13 Discretization -> "FVMup", ... ],14 Scalar[ D[cK[i][z,t],t] == u D[cK[i][z,t],z] +

15 3 beta (cB[i][z,t] - cP[i][z,t])/(Rp (epsP+epsA)),

16 UpperBound -> cK[i][z,t] == cKin[i][t],

17 Discretization -> "FVMdown", ... ],

18 ... },

19 {{i,1,NC-1}},

20 Comment -> "loop for all components"

21 ],... }]];

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Both equations are defined on the common spatial domain Grid1[z], see line

3. The PDE definitions in lines 7-18 are part of the Tensor[...] object in lines

6-21. A Tensor object represents loops over equations. The index variable and

the range for this loop over components are defined in line 19. For a stable MOLdiscretization, the reverse flow directions of the liquid and the solid phases have to

be taken into account. Therefore, we apply two different FV schemes named FVMup

(line 13) and FVMdown (line 17) to discretize the two PDE. These MOL parameter

definitions are explained in the next subsection.

13.5.2.3 Definition of MOL Parameters for Symbolic Preprocessing

The MOL parameters are defined by the MDS functions Domains[...] and

Discretizations[...]. In theDomains function of thecounter-current MBCmodel, only one spatial grid is used for both phases. The reverse flow directions are

considered by two discretization schemes which refer to the common grid function.

This is shown in the Discretizations extract of the MDS definition.

The following MDS example for the grid function of the spatial domain used in the

MBC model is very similar to the definition of the CLR example. Only the attribute

GridPractice is introduced here in line 5. It is an optional attribute for use of FV

methods to define the reference of the grid function (see Section 13.3.2).

1 Domains[

2 Grid[ Grid1[z],

3 Computation -> Function[{k},zl+(k-1)*(zr-zl)/(NE-1)],

4 Granularity -> "NE",

5 GridPractice -> "GridPoints" ]];

The Discretizations object contains two FV schemes defined by the MDS

expressionFVMethod[...]. The FVmethodrequires only the choice of profileas-

sumptions for the 1st and 2nd order spatial derivatives. Therefore, the MDS providesthe attribute Profile.

1 Discretizations[

2 FVMethod[ FVMup,

3 Profile -> {"upwind","piecewise-linear"}],

4 FVMethod[ FVMdown,

5 Profile -> {"downwind","piecewise-linear"} ]];

The first FVMethod[...] is called FVMup. For the 1st order spatial derivative,the upwind profile assumption is defined in line 3. The second FVMethod[...]

shows a downwind profile assumption due to the reverse flow direction of the solid

phase.

13.5.2.4 Simulation Results

The FV schemes presented above have been applied on 150 uniform CVs for MOL

discretization. Each PDE model of a MBC unit with 3 components plus solvent

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consists of 6 PDE, 9 BC, and 3 algebraic equations. They are transformed into 1350

DAE bythesymbolicpreprocessingtool. Simulation resultsareshowninFigure13.14

for the separation of a three component mixture of C8 aromatics. Figure 13.14 shows

typical stationary concentration profiles of the bulk concentrations cB,i , i = 1(1)3.Due to the strong nonlinear behavior of the MBC process, steady-state simulation and

optimization plays an important role for the design of moving-bed chromatographic

processes. Furthermore, dynamic simulation is an important tool for developing new

strategies for process operation and control. Diva is used as an integrated tool to

study these various aspects of moving-bed chromatographic processes.

0

0.1

0.2

0.3

0.4

0.5

0.6

axial coordinate

  m  e  a  s  u  r  e  o   f  c  o  m  p  o  s   i   t   i  o  n

zone I zone II zone III zone IV

FIGURE 13.14

Simulated concentration profiles cB,i of a C8 aromatics separation along theaxial

coordinates zj  ∈ [zl,j , zr,j ], j  = I , I I , I I I , I V  .

13.6 Conclusions and Perspectives

The automatic MOL discretization of the symbolic preprocessing tool SyPProT

within the simulation environment Diva allows to simulate, analyze, and optimize

PDE and IPDE models with the powerful DAE numerical methods provided byDiva.

The symbolic preprocessing tool comprises a Mathematica data structure for sym-

bolic definition of coupled second order PDE and IPDE as well as Mathematica

modules for their MOL discretization. The configurable discretization methods of 

finite-difference and finite-volume schemes are available on uniform, non-uniform,and continuously moving 1-dimensional grids in order to apply the MOL approach to

PDE and IPDE. The translation of the discretized DAE model into a code-generator

input file allows us to use the simulation environment Diva and its DAE numerical

methods.

In many other numerical tools, the MOL approach is implemented without sepa-

ration of model equations and discretization schemes. The model equations are part

of a discretization routine for specific structured problems or they are directly written

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in the discretized form. The application of different discretization schemes leads

to a comparably large effort because the model equations must be reimplemented.

The choice of the appropriate discretization method grows to a major decision within

the complete modeling process. The uniform model representations for symbolicpreprocessing as well as for the generic process unit models of the Diva simulation

kernel reduce the overall effort of modeling and simulation. Model redefinitions are

not necessary either for the application of different MOL discretization schemes to

PDE and IPDE by the symbolic preprocessing tool SyPProT or by using the various

DAE numerical methods of Diva.

Future work in Diva concerning MOL discretization is the implementation of 

high-resolution schemes for hyperbolic partial integro differential equations as used

for modeling of population systems. A promising scheme is the robust upwindscheme [1]. Very interesting is the coupling of high-resolution schemes with the

already implemented moving grid method. Encouraging results obtained by this

approach are described in [20].

Acknowledgments

Main parts of the development of the simulation environment Diva and the pre-

processing tool SyPProT have been funded by Deutsche Forschungsgemeinschaft

(SFB 412). Moreover, many former colleagues and students have been engaged in

this project which is also acknowledged by the authors. Especially the unpublished

student thesis of J. Rieber is emphasized, in the frame of which the moving grid

module has been successfully implemented in SyPProT.

References

[1] M. Brahmadatta, R. Köhler, A. Mitrovic, E.D. Gilles, and M. Zeitz, Symbolic

discretization of population models for process simulation, in S. Pierucci, ed.,

 European Symposiumon ComputerAided Process Engineering- 10, Computer-

Aided Chemical Engineering 8, pages 547–552. Elsevier, May 2000.

[2] K. E. Brenan, S.L. Campbell, and L.R. Petzold, Numerical Solution of Initial

Value Problems in Differential-Algebraic Equations, North Holland, Elsevier

Science Publishing Co. Inc., 1989.

[3] M. Caracotsios andW.E.Stewart, SensitivityAnalysis of Initial Value Problems

with Mixed ODEs and Algebraic Equations, Comp. Chem. Engng., 9, (1985),

359–365.

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[4] P. Deuflhard, E. Hairer, and J. Zugck, One-step and Extrapolation Methods for

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[6] R.M. Furzeland, J.G. Verwer, and P.A. Zegeling, A numerical study of three

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[8] A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comp.

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[9] Harwell Laboratories, Oxford, England, Harwell Library, 1996.

[10] A. Helget and E.D. Gilles, Dynamische Prozeß — und Anlagensimulation, in

H. Schuler, ed., Prozeßsimulation, pages 109–148, VCH, 1994.

[11] P. Holl, W. Marquardt, and E.D. Gilles, DIVA — A powerful tool for dynamic

process simulation, Comp. Chem. Engng., 12, (1988), 421–425.

[12] X. Hua, M. Mangold, A. Kienle, and E.D. Gilles, State profile estimation of an

autothermal periodic fixed-bed reactor, Chem. Engng. Sci., 53, (1998), 47–58.

[13] A. Kienle, G. Lauschke, V. Gehrke, and E.D. Gilles, On the dynamics of the

circulation loop reactor — Numerical methods and analysis, Chem. Engng. Sci.,

50, (1995), 2361–2375.

[14] R. Köhler, A. Gerstlauer, and M. Zeitz, Symbolic preprocessing for simulation

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special issue “Method of Lines,” 2000 (submitted).

[15] R. Köhler, S. Räumschüssel, and M. Zeitz, Code generator for implementing

differential algebraic models used in the process simulation tool DIVA, in A.

Sydow, ed., 15th IMACS World Congress, volume 3, pages 621–626, Berlin,

Germany, 1997. Wissenschaft und Technik Verlag.

[16] B. Koren, A robust upwind discretization method for advection, diffusion and

source terms, in C.B. Vreugdenhiland B. Koren, ed., Numerical Methods for  Advection-Diffusion Problems, pages 117–138. Vieweg, 1993.

[17] A. Kröner, P. Holl, W. Marquardt, and E.D. Gilles, DIVA — An open architec-

ture for dynamic simulation, Comp. Chem. Engng., 14, (1990), 1289–1295.

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[19] B. Van Leer, Upwind-difference methods for aerodynamic problems governed

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differential equations, SIAM J. Sci. Comput., 20, (1998), 719–738.

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[23] M. Mangold, A. Kienle, E.D. Gilles, M. Richter, and E. Roschka, Coupledreaction zones in a circulation loop reactor, Chem. Engng. Sci., 54, (1999),

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